<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c1">S</font> be  <a href="osalg_4.html#V2" title="OSALG_4:attr.2">locally_directed</a> <a href="osalg_1.html#NM3" title="OSALG_1:NM.3">OrderSortedSign</a>;<br/><b>let </b><font color="Maroon" title="c2">X</font> be  <a href="relat_1.html#V2" title="RELAT_1:attr.2">V5</a> <a href="struct_0.html#NM8" title="STRUCT_0:NM.8">ManySortedSet</a> of <font color="Maroon" title="c1">S</font>;<br/><b>let </b><font color="Maroon" title="c3">R</font> be   <a href="msualg_4.html#NM4" title="MSUALG_4:NM.4">ManySortedRelation</a> of <span class="p1">(<span class="default"><a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</a> <font color="Maroon" title="c2">X</font></span>)</span>;<br/>





<b>set </b><font color="Maroon" title="c4">PTA</font> =  <a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</a> <font color="Maroon" title="c2">X</font>;<br/>
<b>set </b><font color="Maroon" title="c5">SPTA</font> = the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</a> <font color="Maroon" title="c2">X</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c6">D</font> =  <a href="osafree.html#K2" title="OSAFREE:func.2">DTConOSA</a> <font color="Maroon" title="c2">X</font>;<br/>
<b>set </b><font color="Maroon" title="c7">OU</font> = <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">OperSymbols</a> of <font color="Maroon" title="c1">S</font>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p2">(<span class="default"><a href="msafree.html#K3" title="MSAFREE:func.3">coprod</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c8">C</font> =  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <span class="p3">(<span class="default"><a href="osafree.html#K2" title="OSAFREE:func.2">DTConOSA</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>;<br/>
<b>set </b><font color="Maroon" title="c9">F</font> =  <a href="osafree.html#K19" title="OSAFREE:func.19">PTClasses</a> <font color="Maroon" title="c2">X</font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font>] <b>means </b>for <font color="Olive" title="b1">s1</font>, <font color="Olive" title="b2">s2</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font><br/> for <font color="Olive" title="b3">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">s1</font> holds <br/>( ( <font color="Olive" title="b1">s1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b2">s2</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="trees_4.html#K1" title="TREES_4:func.1">root-tree</a> <span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">x</font>,<font color="Olive" title="b1">s1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span>)</span>,<span class="p2">(<span class="default"><a href="trees_4.html#K1" title="TREES_4:func.1">root-tree</a> <span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">x</font>,<font color="Olive" title="b1">s1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> $1 <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">s2</font> ) &amp; ( for <font color="Olive" title="b4">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="trees_4.html#K1" title="TREES_4:func.1">root-tree</a> <span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">x</font>,<font color="Olive" title="b1">s1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span>)</span>,<font color="Olive" title="b4">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> $1 <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">s2</font> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b4">y</font>,<span class="p2">(<span class="default"><a href="trees_4.html#K1" title="TREES_4:func.1">root-tree</a> <span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">x</font>,<font color="Olive" title="b1">s1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> $1 <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">s2</font> ) holds <br/>( <font color="Olive" title="b1">s1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b2">s2</font> &amp; <font color="Olive" title="b4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="trees_4.html#K1" title="TREES_4:func.1">root-tree</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">x</font>,<font color="Olive" title="b1">s1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> ) ) );<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font>] <b>means </b>for <font color="Olive" title="b1">o1</font>, <font color="Olive" title="b2">o2</font> being  <a href="osalg_1.html#NM2" title="OSALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">S</font><br/> for <font color="Olive" title="b3">x1</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Olive" title="b1">o1</font>,<span class="p1">(<span class="default"><a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</a> <font color="Maroon" title="c2">X</font></span>)</span><br/> for <font color="Olive" title="b4">x2</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Olive" title="b2">o2</font>,<span class="p1">(<span class="default"><a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</a> <font color="Maroon" title="c2">X</font></span>)</span><br/> for <font color="Olive" title="b5">s3</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font> holds <br/> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Olive" title="b1">o1</font>,<span class="p4">(<span class="default"><a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">x1</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Olive" title="b2">o2</font>,<span class="p4">(<span class="default"><a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b4">x2</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> $1 <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">s3</font> iff ( <font color="Olive" title="b1">o1</font> <a href="osalg_1.html#R1" title="OSALG_1:pred.1">~=</a> <font color="Olive" title="b2">o2</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Olive" title="b1">o1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Olive" title="b2">o2</font></span>)</span> &amp;  <a href="osalg_1.html#K1" title="OSALG_1:func.1">the_result_sort_of</a> <font color="Olive" title="b1">o1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b5">s3</font> &amp;  <a href="osalg_1.html#K1" title="OSALG_1:func.1">the_result_sort_of</a> <font color="Olive" title="b2">o2</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b5">s3</font> &amp;  ex <font color="Olive" title="b6">w3</font> being   <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  st <br/>(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Olive" title="b6">w3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Olive" title="b3">x1</font> &amp; ( for <font color="Olive" title="b7">y</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Olive" title="b6">w3</font> holds <br/><span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b3">x1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b7">y</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b4">x2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b7">y</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> $1 <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b6">w3</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b7">y</font></span>)</span> ) ) ) );<br/>
<a NAME="E1:58"/><i><font color="Green" title="E39">A1</font></i>: 
for <font color="Olive" title="b1">R1</font>, <font color="Olive" title="b2">R2</font> being  <a href="msualg_4.html#NM4" title="MSUALG_4:NM.4">ManySortedRelation</a> of <span class="p1">(<span class="default"><a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</a> <font color="Maroon" title="c2">X</font></span>)</span>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">R1</font>] &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">R1</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">R2</font>] &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b2">R2</font>] holds <br/><font color="Olive" title="b1">R1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Olive" title="b2">R2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">R1</font>, <font color="Maroon" title="c11">R2</font> be   <a href="msualg_4.html#NM4" title="MSUALG_4:NM.4">ManySortedRelation</a> of <span class="p1">(<span class="default"><a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</a> <font color="Maroon" title="c2">X</font></span>)</span>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:58_1"/><i><font color="Green" title="E40">A2</font></i>: 
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c10">R1</font>] &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c10">R1</font>] )
 <b>and </b><br/><a NAME="E2:58_1"/><i><font color="Green" title="E41">A3</font></i>: 
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c11">R2</font>] &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c11">R2</font>] )
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/> for <font color="Olive" title="b2">s</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font> holds <br/> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">$1,<font color="Olive" title="b1">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">R1</font> <a href="msualg_4.html#K1" title="MSUALG_4:func.1">.</a> <font color="Olive" title="b2">s</font> iff <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">$1,<font color="Olive" title="b1">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">R2</font> <a href="msualg_4.html#K1" title="MSUALG_4:func.1">.</a> <font color="Olive" title="b2">s</font> );<br/>
<a NAME="E3:58_1"/><i><font color="Green" title="E42">A4</font></i>: 
for <font color="Olive" title="b1">s</font> being  <a href="lang1.html#NM1" title="LANG1:NM.1">Symbol</a> of <span class="p1">(<span class="default"><a href="osafree.html#K2" title="OSAFREE:func.2">DTConOSA</a> <font color="Maroon" title="c2">X</font></span>)</span>  st <font color="Olive" title="b1">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="dtconstr.html#K6" title="DTCONSTR:func.6">Terminals</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" title="OSAFREE:func.2">DTConOSA</a> <font color="Maroon" title="c2">X</font></span>)</span> holds <br/><font color="Maroon">S<sub>3</sub></font>[ <a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <font color="Olive" title="b1">s</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c12">sy</font> be   <a href="lang1.html#NM1" title="LANG1:NM.1">Symbol</a> of <span class="p1">(<span class="default"><a href="osafree.html#K2" title="OSAFREE:func.2">DTConOSA</a> <font color="Maroon" title="c2">X</font></span>)</span>;<br/>

<b>assume </b><a NAME="E1:58_1_1"/><i><font color="Green" title="E43">A5</font></i>: 
<font color="Maroon" title="c12">sy</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="dtconstr.html#K6" title="DTCONSTR:func.6">Terminals</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" title="OSAFREE:func.2">DTConOSA</a> <font color="Maroon" title="c2">X</font></span>)</span>
 ;<br/>

<b>consider </b><font color="Maroon" title="c13">s</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font>, <font color="Maroon" title="c14">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:58_1_1"/><i><font color="Green" title="E44">A6</font></i>: 
( <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">s</font> &amp; <font color="Maroon" title="c12">sy</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">x</font>,<font color="Maroon" title="c13">s</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <b>by </b><i><a class="txt" href="osafree.html#E1:58_1_1"><i><font color="Green" title="E43">A5</font></i></a>, <a class="ref" href="osafree.html#T4" target="_self" title="OSAFREE:th.4">Th4</a></i>;<br/>
<b>let </b><font color="Maroon" title="c15">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c16">s1</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font>;<br/>



<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:58_1_1_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <font color="Maroon" title="c12">sy</font></span>)</span>,<font color="Maroon" title="c15">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">R1</font> <a href="msualg_4.html#K1" title="MSUALG_4:func.1">.</a> <font color="Maroon" title="c16">s1</font>
 ;<br/><a NAME="E2:58_1_1_1"/><b>then </b>
( <font color="Maroon" title="c13">s</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c16">s1</font> &amp; <font color="Maroon" title="c15">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="trees_4.html#K1" title="TREES_4:func.1">root-tree</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">x</font>,<font color="Maroon" title="c13">s</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <b>by </b><i><a class="txt" href="osafree.html#E1:58_1"><i><font color="Green" title="E40">A2</font></i></a>, <a class="txt" href="osafree.html#E3:58_1_1"><i><font color="Green" title="E44">A6</font></i></a></i>;<br/><b>hence </b><a NAME="E3:58_1_1_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <font color="Maroon" title="c12">sy</font></span>)</span>,<font color="Maroon" title="c15">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">R2</font> <a href="msualg_4.html#K1" title="MSUALG_4:func.1">.</a> <font color="Maroon" title="c16">s1</font>
 <b>by </b><i><a class="txt" href="osafree.html#E2:58_1"><i><font color="Green" title="E41">A3</font></i></a>, <a class="txt" href="osafree.html#E3:58_1_1"><i><font color="Green" title="E44">A6</font></i></a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E5:58_1_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <font color="Maroon" title="c12">sy</font></span>)</span>,<font color="Maroon" title="c15">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">R2</font> <a href="msualg_4.html#K1" title="MSUALG_4:func.1">.</a> <font color="Maroon" title="c16">s1</font>
 ;<br/>

<a NAME="E6:58_1_1"/><b>then </b>
( <font color="Maroon" title="c13">s</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c16">s1</font> &amp; <font color="Maroon" title="c15">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="trees_4.html#K1" title="TREES_4:func.1">root-tree</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">x</font>,<font color="Maroon" title="c13">s</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 
<b>by </b><i><a class="txt" href="osafree.html#E2:58_1"><i><font color="Green" title="E41">A3</font></i></a>, <a class="txt" href="osafree.html#E3:58_1_1"><i><font color="Green" title="E44">A6</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:58_1_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <font color="Maroon" title="c12">sy</font></span>)</span>,<font color="Maroon" title="c15">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">R1</font> <a href="msualg_4.html#K1" title="MSUALG_4:func.1">.</a> <font color="Maroon" title="c16">s1</font>
 <b>by </b><i><a class="txt" href="osafree.html#E1:58_1"><i><font color="Green" title="E40">A2</font></i></a>, <a class="txt" href="osafree.html#E3:58_1_1"><i><font color="Green" title="E44">A6</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:58_1"/><i><font color="Green" title="E43">A7</font></i>: 
for <font color="Olive" title="b1">nt</font> being  <a href="lang1.html#NM1" title="LANG1:NM.1">Symbol</a> of <span class="p1">(<span class="default"><a href="osafree.html#K2" title="OSAFREE:func.2">DTConOSA</a> <font color="Maroon" title="c2">X</font></span>)</span><br/> for <font color="Olive" title="b2">ts</font> being   <a href="trees_4.html#M1" title="TREES_4:mode.1">FinSequence</a> of  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" title="OSAFREE:func.2">DTConOSA</a> <font color="Maroon" title="c2">X</font></span>)</span>  st <font color="Olive" title="b1">nt</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a>  <a href="dtconstr.html#K1" title="DTCONSTR:func.1">roots</a> <font color="Olive" title="b2">ts</font> &amp; ( for <font color="Olive" title="b3">t</font> being  <a href="trees_2.html#NM6" title="TREES_2:NM.6">DecoratedTree</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="osafree.html#K2" title="OSAFREE:func.2">DTConOSA</a> <font color="Maroon" title="c2">X</font></span>)</span>  st <font color="Olive" title="b3">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b2">ts</font> holds <br/><font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b3">t</font>] ) holds <br/><font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">nt</font> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Olive" title="b2">ts</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c12">nt</font> be   <a href="lang1.html#NM1" title="LANG1:NM.1">Symbol</a> of <span class="p1">(<span class="default"><a href="osafree.html#K2" title="OSAFREE:func.2">DTConOSA</a> <font color="Maroon" title="c2">X</font></span>)</span>;<br/><b>let </b><font color="Maroon" title="c13">ts</font> be    <a href="trees_4.html#M1" title="TREES_4:mode.1">FinSequence</a> of  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" title="OSAFREE:func.2">DTConOSA</a> <font color="Maroon" title="c2">X</font></span>)</span>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:58_1_2"/><i><font color="Green" title="E44">A8</font></i>: 
<font color="Maroon" title="c12">nt</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a>  <a href="dtconstr.html#K1" title="DTCONSTR:func.1">roots</a> <font color="Maroon" title="c13">ts</font>
 <b>and </b><br/><a NAME="E2:58_1_2"/><i><font color="Green" title="E45">A9</font></i>: 
for <font color="Olive" title="b1">t</font> being  <a href="trees_2.html#NM6" title="TREES_2:NM.6">DecoratedTree</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="osafree.html#K2" title="OSAFREE:func.2">DTConOSA</a> <font color="Maroon" title="c2">X</font></span>)</span>  st <font color="Olive" title="b1">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c13">ts</font> holds <br/><font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">t</font>]
 ;<br/>

<a NAME="E3:58_1_2"/>
<font color="Maroon" title="c12">nt</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">s</font> where <font color="Olive" title="b1">s</font> is   <a href="lang1.html#NM1" title="LANG1:NM.1">Symbol</a> of <span class="p1">(<span class="default"><a href="osafree.html#K2" title="OSAFREE:func.2">DTConOSA</a> <font color="Maroon" title="c2">X</font></span>)</span> :  ex <font color="Olive" title="b2">n</font> being  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> st <font color="Olive" title="b1">s</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a> <font color="Olive" title="b2">n</font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="osafree.html#E1:58_1_2"><i><font color="Green" title="E44">A8</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c14">nt1</font> = <font color="Maroon" title="c12">nt</font> as   <a href="dtconstr.html#NM3" title="DTCONSTR:NM.3">NonTerminal</a> of <span class="p1">(<span class="default"><a href="osafree.html#K2" title="OSAFREE:func.2">DTConOSA</a> <font color="Maroon" title="c2">X</font></span>)</span> <b>by </b><i><a class="ref" href="lang1.html#D3" title="LANG1:def.3">LANG1:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c15">tss</font> = <font color="Maroon" title="c13">ts</font> as    <a href="dtconstr.html#M2" title="DTCONSTR:mode.2">SubtreeSeq</a> of <font color="Maroon" title="c14">nt1</font> <b>by </b><i><a class="txt" href="osafree.html#E1:58_1_2"><i><font color="Green" title="E44">A8</font></i></a>, <a class="ref" href="dtconstr.html#D9" title="DTCONSTR:def.9">DTCONSTR:def 9</a></i>;<br/>
<b>let </b><font color="Maroon" title="c16">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c17">s</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font>;<br/>



<i><font color="Green" title="E46">A10</font></i>: the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">s</font> = 
 <a href="osafree.html#K4" title="OSAFREE:func.4">ParsedTerms</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c17">s</font>
<b>by </b><i><a class="ref" href="osafree.html#D8" target="_self" title="OSAFREE:def.8">Def8</a></i>
<br/>.= 
<span class="p1">{<span class="default"> <font color="Olive" title="b1">a1</font> where <font color="Olive" title="b1">a1</font> is    <a href="dtconstr.html#M1" title="DTCONSTR:mode.1">Element</a> of  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" title="OSAFREE:func.2">DTConOSA</a> <font color="Maroon" title="c2">X</font></span>)</span> : (  ex <font color="Olive" title="b2">s1</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font> ex <font color="Olive" title="b3">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">s1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c17">s</font> &amp; <font color="Olive" title="b3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">s1</font> &amp; <font color="Olive" title="b1">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="trees_4.html#K1" title="TREES_4:func.1">root-tree</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">x</font>,<font color="Olive" title="b2">s1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> ) or  ex <font color="Olive" title="b2">o</font> being  <a href="osalg_1.html#NM2" title="OSALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">S</font> st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">o</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp;  <a href="osalg_1.html#K1" title="OSALG_1:func.1">the_result_sort_of</a> <font color="Olive" title="b2">o</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c17">s</font> ) ) </span>}</span> 

;<br/>
<a NAME="E7:58_1_2"/><i><font color="Green" title="E47">A11</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">nt</font>,<span class="p2">(<span class="default"><a href="dtconstr.html#K1" title="DTCONSTR:func.1">roots</a> <font color="Maroon" title="c13">ts</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="lang1.html#U1" title="LANG1:sel.1">Rules</a> of <span class="p1">(<span class="default"><a href="osafree.html#K2" title="OSAFREE:func.2">DTConOSA</a> <font color="Maroon" title="c2">X</font></span>)</span>
 
<b>by </b><i><a class="txt" href="osafree.html#E1:58_1_2"><i><font color="Green" title="E44">A8</font></i></a>, <a class="ref" href="lang1.html#D1" title="LANG1:def.1">LANG1:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c18">sy</font> = <font color="Maroon" title="c12">nt</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">OperSymbols</a> of <font color="Maroon" title="c1">S</font>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p2">(<span class="default"><a href="msafree.html#K3" title="MSAFREE:func.3">coprod</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span> ;<br/>
<b>reconsider </b><font color="Maroon" title="c19">rt</font> =  <a href="dtconstr.html#K1" title="DTCONSTR:func.1">roots</a> <font color="Maroon" title="c13">ts</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <span class="p1">(<span class="default"><span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">OperSymbols</a> of <font color="Maroon" title="c1">S</font>,<span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2">(<span class="default"><a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p3">(<span class="default"><a href="msafree.html#K3" title="MSAFREE:func.3">coprod</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span></span>)</span> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  <b>by </b><i><a class="txt" href="osafree.html#E7:58_1_2"><i><font color="Green" title="E47">A11</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E10:58_1_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">sy</font>,<font color="Maroon" title="c19">rt</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="osafree.html#K1" title="OSAFREE:func.1">OSREL</a> <font color="Maroon" title="c2">X</font>
 
<b>by </b><i><a class="txt" href="osafree.html#E1:58_1_2"><i><font color="Green" title="E44">A8</font></i></a>, <a class="ref" href="lang1.html#D1" title="LANG1:def.1">LANG1:def 1</a></i>;<br/>
<a NAME="E11:58_1_2"/><b>then </b>
<font color="Maroon" title="c18">sy</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">OperSymbols</a> of <font color="Maroon" title="c1">S</font>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="osafree.html#D4" target="_self" title="OSAFREE:def.4">Def4</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c20">o</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">OperSymbols</a> of <font color="Maroon" title="c1">S</font>, <font color="Maroon" title="c21">x2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span><b> such that </b><br/><a NAME="E13:58_1_2"/><i><font color="Green" title="E48">A12</font></i>: 
<font color="Maroon" title="c18">sy</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c20">o</font>,<font color="Maroon" title="c21">x2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="ref" href="domain_1.html#T9" title="DOMAIN_1:th.9">DOMAIN_1:9</a></i>;<br/>
<a NAME="E14:58_1_2"/><i><font color="Green" title="E49">A13</font></i>: 
<font color="Maroon" title="c21">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E15:58_1_2"/><b>then </b><i><font color="Green" title="E50">A14</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">nt</font> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Maroon" title="c15">tss</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c20">o</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<b>by </b><i><a class="txt" href="osafree.html#E13:58_1_2"><i><font color="Green" title="E48">A12</font></i></a>, <a class="ref" href="trees_4.html#D4" title="TREES_4:def.4">TREES_4:def 4</a></i>;<br/>
<a NAME="E16:58_1_2"/><i><font color="Green" title="E51">A15</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c13">ts</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" title="OSAFREE:func.2">DTConOSA</a> <font color="Maroon" title="c2">X</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></i>;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:58_1_2_2"/><i><font color="Green" title="E52">A16</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">nt</font> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Maroon" title="c13">ts</font></span>)</span>,<font color="Maroon" title="c16">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">R1</font> <a href="msualg_4.html#K1" title="MSUALG_4:func.1">.</a> <font color="Maroon" title="c17">s</font>
 ;<br/><a NAME="E2:58_1_2_2"/><b>then </b>
( <font color="Maroon" title="c12">nt</font> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Maroon" title="c13">ts</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">s</font> &amp; <font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">s</font> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><b>then consider </b><font color="Maroon" title="c22">a1</font> being    <a href="dtconstr.html#M1" title="DTCONSTR:mode.1">Element</a> of  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" title="OSAFREE:func.2">DTConOSA</a> <font color="Maroon" title="c2">X</font></span>)</span><b> such that </b><br/><a NAME="E4:58_1_2_2"/><i><font color="Green" title="E53">A17</font></i>: 
<font color="Maroon" title="c16">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">a1</font>
 <b>and </b><br/><a NAME="E5:58_1_2_2"/><i><font color="Green" title="E54">A18</font></i>: 
(  ex <font color="Olive" title="b1">s1</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font> ex <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">s1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c17">s</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">s1</font> &amp; <font color="Maroon" title="c22">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="trees_4.html#K1" title="TREES_4:func.1">root-tree</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">y</font>,<font color="Olive" title="b1">s1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> ) or  ex <font color="Olive" title="b1">o</font> being  <a href="osalg_1.html#NM2" title="OSALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">S</font> st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">o</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">a1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp;  <a href="osalg_1.html#K1" title="OSALG_1:func.1">the_result_sort_of</a> <font color="Olive" title="b1">o</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c17">s</font> ) )
 <b>by </b><i><a class="txt" href="osafree.html#E6:58_1_2"><i><font color="Green" title="E46">A10</font></i></a></i>;<br/><a NAME="E6:58_1_2_2"/>
for <font color="Olive" title="b1">s1</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font><br/> for <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Olive" title="b1">s1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c17">s</font> or  not <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">s1</font> or  not <font color="Maroon" title="c22">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="trees_4.html#K1" title="TREES_4:func.1">root-tree</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">y</font>,<font color="Olive" title="b1">s1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_2_2_1"><b>proof </b></a><div class="add">

<b>given </b><font color="Maroon" title="c23">s1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font>, <font color="Maroon" title="c24">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><a NAME="E1:58_1_2_2_1"/><i><font color="Green" title="E55">A19</font></i>: 
( <font color="Maroon" title="c23">s1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c17">s</font> &amp; <font color="Maroon" title="c24">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">s1</font> &amp; <font color="Maroon" title="c22">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="trees_4.html#K1" title="TREES_4:func.1">root-tree</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c24">y</font>,<font color="Maroon" title="c23">s1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 ;<br/>

<a NAME="E2:58_1_2_2_1"/>
<font color="Maroon" title="c12">nt</font> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Maroon" title="c13">ts</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="trees_4.html#K1" title="TREES_4:func.1">root-tree</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c24">y</font>,<font color="Maroon" title="c23">s1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<b>by </b><i><a class="txt" href="osafree.html#E1:58_1"><i><font color="Green" title="E40">A2</font></i></a>, <a class="txt" href="osafree.html#E1:58_1_2_2"><i><font color="Green" title="E52">A16</font></i></a>, <a class="txt" href="osafree.html#E4:58_1_2_2"><i><font color="Green" title="E53">A17</font></i></a>, <a class="txt" href="osafree.html#E1:58_1_2_2_1"><i><font color="Green" title="E55">A19</font></i></a></i>;<br/>
<a NAME="E3:58_1_2_2_1"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c24">y</font>,<font color="Maroon" title="c23">s1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">nt</font> &amp; <font color="Maroon" title="c13">ts</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 
<b>by </b><i><a class="ref" href="trees_4.html#T17" title="TREES_4:th.17">TREES_4:17</a></i>;<br/>
<a NAME="E4:58_1_2_2_1"/><b>then </b>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">s1</font>
 
<b>by </b><i><a class="txt" href="osafree.html#E13:58_1_2"><i><font color="Green" title="E48">A12</font></i></a>, <a class="txt" href="osafree.html#E14:58_1_2"><i><font color="Green" title="E49">A13</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/>
<a NAME="E5:58_1_2_2_1"/><b>then </b>
<font color="Maroon" title="c23">s1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c23">s1</font>
 
;<br/>
<b>hence </b><a NAME="E6:58_1_2_2_1"/>
contradiction
 ;<br/>


</div><b>end;</b></div><b>then consider </b><font color="Maroon" title="c23">o1</font> being   <a href="osalg_1.html#NM2" title="OSALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">S</font><b> such that </b><br/><a NAME="E8:58_1_2_2"/><i><font color="Green" title="E55">A20</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c23">o1</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">a1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp;  <a href="osalg_1.html#K1" title="OSALG_1:func.1">the_result_sort_of</a> <font color="Maroon" title="c23">o1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c17">s</font> )
 <b>by </b><i><a class="txt" href="osafree.html#E5:58_1_2_2"><i><font color="Green" title="E54">A18</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c24">ts1</font> being    <a href="dtconstr.html#M2" title="DTCONSTR:mode.2">SubtreeSeq</a> of  <a href="osafree.html#K9" title="OSAFREE:func.9">OSSym</a> <font color="Maroon" title="c23">o1</font>,<font color="Maroon" title="c2">X</font><b> such that </b><br/><a NAME="E10:58_1_2_2"/><i><font color="Green" title="E56">A21</font></i>: 
( <font color="Maroon" title="c22">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="osafree.html#K9" title="OSAFREE:func.9">OSSym</a> <font color="Maroon" title="c23">o1</font>,<font color="Maroon" title="c2">X</font></span>)</span> <a href="dtconstr.html#K9" title="DTCONSTR:func.9">-tree</a> <font color="Maroon" title="c24">ts1</font> &amp;  <a href="osafree.html#K9" title="OSAFREE:func.9">OSSym</a> <font color="Maroon" title="c23">o1</font>,<font color="Maroon" title="c2">X</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a>  <a href="dtconstr.html#K1" title="DTCONSTR:func.1">roots</a> <font color="Maroon" title="c24">ts1</font> &amp; <font color="Maroon" title="c24">ts1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Maroon" title="c23">o1</font>,<span class="p1">(<span class="default"><a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</a> <font color="Maroon" title="c2">X</font></span>)</span> &amp; <font color="Maroon" title="c22">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c23">o1</font>,<span class="p2">(<span class="default"><a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">ts1</font> )
 <b>by </b><i><a class="txt" href="osafree.html#E8:58_1_2_2"><i><font color="Green" title="E55">A20</font></i></a>, <a class="ref" href="osafree.html#T11" target="_self" title="OSAFREE:th.11">Th11</a></i>;<br/><b>consider </b><font color="Maroon" title="c25">ts2</font> being    <a href="dtconstr.html#M2" title="DTCONSTR:mode.2">SubtreeSeq</a> of  <a href="osafree.html#K9" title="OSAFREE:func.9">OSSym</a> <font color="Maroon" title="c20">o</font>,<font color="Maroon" title="c2">X</font><b> such that </b><br/><a NAME="E12:58_1_2_2"/><i><font color="Green" title="E57">A22</font></i>: 
( <font color="Maroon" title="c14">nt1</font> <a href="dtconstr.html#K9" title="DTCONSTR:func.9">-tree</a> <font color="Maroon" title="c15">tss</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="osafree.html#K9" title="OSAFREE:func.9">OSSym</a> <font color="Maroon" title="c20">o</font>,<font color="Maroon" title="c2">X</font></span>)</span> <a href="dtconstr.html#K9" title="DTCONSTR:func.9">-tree</a> <font color="Maroon" title="c25">ts2</font> &amp;  <a href="osafree.html#K9" title="OSAFREE:func.9">OSSym</a> <font color="Maroon" title="c20">o</font>,<font color="Maroon" title="c2">X</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a>  <a href="dtconstr.html#K1" title="DTCONSTR:func.1">roots</a> <font color="Maroon" title="c25">ts2</font> &amp; <font color="Maroon" title="c25">ts2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Maroon" title="c20">o</font>,<span class="p1">(<span class="default"><a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</a> <font color="Maroon" title="c2">X</font></span>)</span> &amp; <font color="Maroon" title="c14">nt1</font> <a href="dtconstr.html#K9" title="DTCONSTR:func.9">-tree</a> <font color="Maroon" title="c15">tss</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c20">o</font>,<span class="p2">(<span class="default"><a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">ts2</font> )
 <b>by </b><i><a class="txt" href="osafree.html#E15:58_1_2"><i><font color="Green" title="E50">A14</font></i></a>, <a class="ref" href="osafree.html#T11" target="_self" title="OSAFREE:th.11">Th11</a></i>;<br/><a NAME="E13:58_1_2_2"/><i><font color="Green" title="E58">A23</font></i>: 
<font color="Maroon" title="c25">ts2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">tss</font>
 <b>by </b><i><a class="txt" href="osafree.html#E12:58_1_2_2"><i><font color="Green" title="E57">A22</font></i></a>, <a class="ref" href="trees_4.html#T15" title="TREES_4:th.15">TREES_4:15</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c26">tsa</font> = <font color="Maroon" title="c24">ts1</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Maroon" title="c23">o1</font>,<span class="p1">(<span class="default"><a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</a> <font color="Maroon" title="c2">X</font></span>)</span> <b>by </b><i><a class="txt" href="osafree.html#E10:58_1_2_2"><i><font color="Green" title="E56">A21</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c27">tsb</font> = <font color="Maroon" title="c25">ts2</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Maroon" title="c20">o</font>,<span class="p1">(<span class="default"><a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</a> <font color="Maroon" title="c2">X</font></span>)</span> <b>by </b><i><a class="txt" href="osafree.html#E12:58_1_2_2"><i><font color="Green" title="E57">A22</font></i></a></i>;<br/><a NAME="E16:58_1_2_2"/><i><font color="Green" title="E59">A24</font></i>: 
( <font color="Maroon" title="c20">o</font> <a href="osalg_1.html#R1" title="OSALG_1:pred.1">~=</a> <font color="Maroon" title="c23">o1</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c20">o</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c23">o1</font></span>)</span> &amp;  <a href="osalg_1.html#K1" title="OSALG_1:func.1">the_result_sort_of</a> <font color="Maroon" title="c20">o</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c17">s</font> &amp;  <a href="osalg_1.html#K1" title="OSALG_1:func.1">the_result_sort_of</a> <font color="Maroon" title="c23">o1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c17">s</font> &amp;  ex <font color="Olive" title="b1">w3</font> being   <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  st <br/>(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Olive" title="b1">w3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c27">tsb</font> &amp; ( for <font color="Olive" title="b2">y</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Olive" title="b1">w3</font> holds <br/><span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c27">tsb</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">y</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c26">tsa</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">y</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">R1</font> <a href="msualg_4.html#K1" title="MSUALG_4:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">w3</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b2">y</font></span>)</span> ) ) )
 <b>by </b><i><a class="txt" href="osafree.html#E1:58_1"><i><font color="Green" title="E40">A2</font></i></a>, <a class="txt" href="osafree.html#E1:58_1_2_2"><i><font color="Green" title="E52">A16</font></i></a>, <a class="txt" href="osafree.html#E4:58_1_2_2"><i><font color="Green" title="E53">A17</font></i></a>, <a class="txt" href="osafree.html#E10:58_1_2_2"><i><font color="Green" title="E56">A21</font></i></a>, <a class="txt" href="osafree.html#E12:58_1_2_2"><i><font color="Green" title="E57">A22</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c28">w3</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> <b> such that </b><br/><a NAME="E18:58_1_2_2"/><i><font color="Green" title="E60">A25</font></i>: 
(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c28">w3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c27">tsb</font> &amp; ( for <font color="Olive" title="b1">y</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c28">w3</font> holds <br/><span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c27">tsb</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">y</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c26">tsa</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">y</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">R1</font> <a href="msualg_4.html#K1" title="MSUALG_4:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c28">w3</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b1">y</font></span>)</span> ) )
 <b>by </b><i><a class="txt" href="osafree.html#E1:58_1"><i><font color="Green" title="E40">A2</font></i></a>, <a class="txt" href="osafree.html#E1:58_1_2_2"><i><font color="Green" title="E52">A16</font></i></a>, <a class="txt" href="osafree.html#E4:58_1_2_2"><i><font color="Green" title="E53">A17</font></i></a>, <a class="txt" href="osafree.html#E10:58_1_2_2"><i><font color="Green" title="E56">A21</font></i></a>, <a class="txt" href="osafree.html#E12:58_1_2_2"><i><font color="Green" title="E57">A22</font></i></a></i>;<br/><a NAME="E19:58_1_2_2"/>
for <font color="Olive" title="b1">y</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c28">w3</font> holds <br/><span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c27">tsb</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">y</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c26">tsa</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">y</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">R2</font> <a href="msualg_4.html#K1" title="MSUALG_4:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c28">w3</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b1">y</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_2_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c29">y</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:58_1_2_2_2"/><i><font color="Green" title="E61">A26</font></i>: 
<font color="Maroon" title="c29">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c28">w3</font>
 ;<br/>

<a NAME="E2:58_1_2_2_2"/><i><font color="Green" title="E62">A27</font></i>: 
<font color="Maroon" title="c27">tsb</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c13">ts</font>
 
<b>by </b><i><a class="txt" href="osafree.html#E13:58_1_2_2"><i><font color="Green" title="E58">A23</font></i></a>, <a class="txt" href="osafree.html#E18:58_1_2_2"><i><font color="Green" title="E60">A25</font></i></a>, <a class="txt" href="osafree.html#E1:58_1_2_2_2"><i><font color="Green" title="E61">A26</font></i></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c30">t</font> = <font color="Maroon" title="c27">tsb</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">y</font> as    <a href="dtconstr.html#M1" title="DTCONSTR:mode.1">Element</a> of  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" title="OSAFREE:func.2">DTConOSA</a> <font color="Maroon" title="c2">X</font></span>)</span> <b>by </b><i><a class="txt" href="osafree.html#E16:58_1_2"><i><font color="Green" title="E51">A15</font></i></a></i>;<br/>
<a NAME="E4:58_1_2_2_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c30">t</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c26">tsa</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">y</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">R1</font> <a href="msualg_4.html#K1" title="MSUALG_4:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c28">w3</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c29">y</font></span>)</span>
 
<b>by </b><i><a class="txt" href="osafree.html#E18:58_1_2_2"><i><font color="Green" title="E60">A25</font></i></a>, <a class="txt" href="osafree.html#E1:58_1_2_2_2"><i><font color="Green" title="E61">A26</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:58_1_2_2_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c27">tsb</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">y</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c26">tsa</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">y</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">R2</font> <a href="msualg_4.html#K1" title="MSUALG_4:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c28">w3</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c29">y</font></span>)</span>
 <b>by </b><i><a class="txt" href="osafree.html#E2:58_1_2"><i><font color="Green" title="E45">A9</font></i></a>, <a class="txt" href="osafree.html#E2:58_1_2_2_2"><i><font color="Green" title="E62">A27</font></i></a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E20:58_1_2_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">nt</font> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Maroon" title="c13">ts</font></span>)</span>,<font color="Maroon" title="c16">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">R2</font> <a href="msualg_4.html#K1" title="MSUALG_4:func.1">.</a> <font color="Maroon" title="c17">s</font>
 <b>by </b><i><a class="txt" href="osafree.html#E2:58_1"><i><font color="Green" title="E41">A3</font></i></a>, <a class="txt" href="osafree.html#E4:58_1_2_2"><i><font color="Green" title="E53">A17</font></i></a>, <a class="txt" href="osafree.html#E10:58_1_2_2"><i><font color="Green" title="E56">A21</font></i></a>, <a class="txt" href="osafree.html#E12:58_1_2_2"><i><font color="Green" title="E57">A22</font></i></a>, <a class="txt" href="osafree.html#E16:58_1_2_2"><i><font color="Green" title="E59">A24</font></i></a>, <a class="txt" href="osafree.html#E18:58_1_2_2"><i><font color="Green" title="E60">A25</font></i></a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E18:58_1_2"/><i><font color="Green" title="E52">A28</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">nt</font> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Maroon" title="c13">ts</font></span>)</span>,<font color="Maroon" title="c16">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">R2</font> <a href="msualg_4.html#K1" title="MSUALG_4:func.1">.</a> <font color="Maroon" title="c17">s</font>
 ;<br/>

<a NAME="E19:58_1_2"/><b>then </b>
( <font color="Maroon" title="c12">nt</font> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Maroon" title="c13">ts</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">s</font> &amp; <font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">s</font> )
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c22">a1</font> being    <a href="dtconstr.html#M1" title="DTCONSTR:mode.1">Element</a> of  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" title="OSAFREE:func.2">DTConOSA</a> <font color="Maroon" title="c2">X</font></span>)</span><b> such that </b><br/><a NAME="E21:58_1_2"/><i><font color="Green" title="E53">A29</font></i>: 
<font color="Maroon" title="c16">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">a1</font>
 <b>and </b><br/><a NAME="E22:58_1_2"/><i><font color="Green" title="E54">A30</font></i>: 
(  ex <font color="Olive" title="b1">s1</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font> ex <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">s1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c17">s</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">s1</font> &amp; <font color="Maroon" title="c22">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="trees_4.html#K1" title="TREES_4:func.1">root-tree</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">y</font>,<font color="Olive" title="b1">s1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> ) or  ex <font color="Olive" title="b1">o</font> being  <a href="osalg_1.html#NM2" title="OSALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">S</font> st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">o</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">a1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp;  <a href="osalg_1.html#K1" title="OSALG_1:func.1">the_result_sort_of</a> <font color="Olive" title="b1">o</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c17">s</font> ) )
 <b>by </b><i><a class="txt" href="osafree.html#E6:58_1_2"><i><font color="Green" title="E46">A10</font></i></a></i>;<br/>
<a NAME="E23:58_1_2"/>
for <font color="Olive" title="b1">s1</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font><br/> for <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Olive" title="b1">s1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c17">s</font> or  not <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">s1</font> or  not <font color="Maroon" title="c22">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="trees_4.html#K1" title="TREES_4:func.1">root-tree</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">y</font>,<font color="Olive" title="b1">s1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_2_3"><b>proof </b></a><div class="add">

<b>given </b><font color="Maroon" title="c23">s1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font>, <font color="Maroon" title="c24">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><a NAME="E1:58_1_2_3"/><i><font color="Green" title="E55">A31</font></i>: 
( <font color="Maroon" title="c23">s1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c17">s</font> &amp; <font color="Maroon" title="c24">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">s1</font> &amp; <font color="Maroon" title="c22">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="trees_4.html#K1" title="TREES_4:func.1">root-tree</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c24">y</font>,<font color="Maroon" title="c23">s1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 ;<br/>

<a NAME="E2:58_1_2_3"/>
<font color="Maroon" title="c12">nt</font> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Maroon" title="c13">ts</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="trees_4.html#K1" title="TREES_4:func.1">root-tree</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c24">y</font>,<font color="Maroon" title="c23">s1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<b>by </b><i><a class="txt" href="osafree.html#E2:58_1"><i><font color="Green" title="E41">A3</font></i></a>, <a class="txt" href="osafree.html#E18:58_1_2"><i><font color="Green" title="E52">A28</font></i></a>, <a class="txt" href="osafree.html#E21:58_1_2"><i><font color="Green" title="E53">A29</font></i></a>, <a class="txt" href="osafree.html#E1:58_1_2_3"><i><font color="Green" title="E55">A31</font></i></a></i>;<br/>
<a NAME="E3:58_1_2_3"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c24">y</font>,<font color="Maroon" title="c23">s1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">nt</font> &amp; <font color="Maroon" title="c13">ts</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 
<b>by </b><i><a class="ref" href="trees_4.html#T17" title="TREES_4:th.17">TREES_4:17</a></i>;<br/>
<a NAME="E4:58_1_2_3"/><b>then </b>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">s1</font>
 
<b>by </b><i><a class="txt" href="osafree.html#E13:58_1_2"><i><font color="Green" title="E48">A12</font></i></a>, <a class="txt" href="osafree.html#E14:58_1_2"><i><font color="Green" title="E49">A13</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/>
<a NAME="E5:58_1_2_3"/><b>then </b>
<font color="Maroon" title="c23">s1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c23">s1</font>
 
;<br/>
<b>hence </b><a NAME="E6:58_1_2_3"/>
contradiction
 ;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon" title="c23">o1</font> being   <a href="osalg_1.html#NM2" title="OSALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">S</font><b> such that </b><br/><a NAME="E25:58_1_2"/><i><font color="Green" title="E55">A32</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c23">o1</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">a1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp;  <a href="osalg_1.html#K1" title="OSALG_1:func.1">the_result_sort_of</a> <font color="Maroon" title="c23">o1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c17">s</font> )
 <b>by </b><i><a class="txt" href="osafree.html#E22:58_1_2"><i><font color="Green" title="E54">A30</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c24">ts1</font> being    <a href="dtconstr.html#M2" title="DTCONSTR:mode.2">SubtreeSeq</a> of  <a href="osafree.html#K9" title="OSAFREE:func.9">OSSym</a> <font color="Maroon" title="c23">o1</font>,<font color="Maroon" title="c2">X</font><b> such that </b><br/><a NAME="E27:58_1_2"/><i><font color="Green" title="E56">A33</font></i>: 
( <font color="Maroon" title="c22">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="osafree.html#K9" title="OSAFREE:func.9">OSSym</a> <font color="Maroon" title="c23">o1</font>,<font color="Maroon" title="c2">X</font></span>)</span> <a href="dtconstr.html#K9" title="DTCONSTR:func.9">-tree</a> <font color="Maroon" title="c24">ts1</font> &amp;  <a href="osafree.html#K9" title="OSAFREE:func.9">OSSym</a> <font color="Maroon" title="c23">o1</font>,<font color="Maroon" title="c2">X</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a>  <a href="dtconstr.html#K1" title="DTCONSTR:func.1">roots</a> <font color="Maroon" title="c24">ts1</font> &amp; <font color="Maroon" title="c24">ts1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Maroon" title="c23">o1</font>,<span class="p1">(<span class="default"><a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</a> <font color="Maroon" title="c2">X</font></span>)</span> &amp; <font color="Maroon" title="c22">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c23">o1</font>,<span class="p2">(<span class="default"><a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">ts1</font> )
 <b>by </b><i><a class="txt" href="osafree.html#E25:58_1_2"><i><font color="Green" title="E55">A32</font></i></a>, <a class="ref" href="osafree.html#T11" target="_self" title="OSAFREE:th.11">Th11</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c25">ts2</font> being    <a href="dtconstr.html#M2" title="DTCONSTR:mode.2">SubtreeSeq</a> of  <a href="osafree.html#K9" title="OSAFREE:func.9">OSSym</a> <font color="Maroon" title="c20">o</font>,<font color="Maroon" title="c2">X</font><b> such that </b><br/><a NAME="E29:58_1_2"/><i><font color="Green" title="E57">A34</font></i>: 
( <font color="Maroon" title="c14">nt1</font> <a href="dtconstr.html#K9" title="DTCONSTR:func.9">-tree</a> <font color="Maroon" title="c15">tss</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="osafree.html#K9" title="OSAFREE:func.9">OSSym</a> <font color="Maroon" title="c20">o</font>,<font color="Maroon" title="c2">X</font></span>)</span> <a href="dtconstr.html#K9" title="DTCONSTR:func.9">-tree</a> <font color="Maroon" title="c25">ts2</font> &amp;  <a href="osafree.html#K9" title="OSAFREE:func.9">OSSym</a> <font color="Maroon" title="c20">o</font>,<font color="Maroon" title="c2">X</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a>  <a href="dtconstr.html#K1" title="DTCONSTR:func.1">roots</a> <font color="Maroon" title="c25">ts2</font> &amp; <font color="Maroon" title="c25">ts2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Maroon" title="c20">o</font>,<span class="p1">(<span class="default"><a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</a> <font color="Maroon" title="c2">X</font></span>)</span> &amp; <font color="Maroon" title="c14">nt1</font> <a href="dtconstr.html#K9" title="DTCONSTR:func.9">-tree</a> <font color="Maroon" title="c15">tss</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c20">o</font>,<span class="p2">(<span class="default"><a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">ts2</font> )
 <b>by </b><i><a class="txt" href="osafree.html#E15:58_1_2"><i><font color="Green" title="E50">A14</font></i></a>, <a class="ref" href="osafree.html#T11" target="_self" title="OSAFREE:th.11">Th11</a></i>;<br/>
<a NAME="E30:58_1_2"/><i><font color="Green" title="E58">A35</font></i>: 
<font color="Maroon" title="c25">ts2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">tss</font>
 
<b>by </b><i><a class="txt" href="osafree.html#E29:58_1_2"><i><font color="Green" title="E57">A34</font></i></a>, <a class="ref" href="trees_4.html#T15" title="TREES_4:th.15">TREES_4:15</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c26">tsa</font> = <font color="Maroon" title="c24">ts1</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Maroon" title="c23">o1</font>,<span class="p1">(<span class="default"><a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</a> <font color="Maroon" title="c2">X</font></span>)</span> <b>by </b><i><a class="txt" href="osafree.html#E27:58_1_2"><i><font color="Green" title="E56">A33</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c27">tsb</font> = <font color="Maroon" title="c25">ts2</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Maroon" title="c20">o</font>,<span class="p1">(<span class="default"><a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</a> <font color="Maroon" title="c2">X</font></span>)</span> <b>by </b><i><a class="txt" href="osafree.html#E29:58_1_2"><i><font color="Green" title="E57">A34</font></i></a></i>;<br/>
<a NAME="E33:58_1_2"/><i><font color="Green" title="E59">A36</font></i>: 
( <font color="Maroon" title="c20">o</font> <a href="osalg_1.html#R1" title="OSALG_1:pred.1">~=</a> <font color="Maroon" title="c23">o1</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c20">o</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c23">o1</font></span>)</span> &amp;  <a href="osalg_1.html#K1" title="OSALG_1:func.1">the_result_sort_of</a> <font color="Maroon" title="c20">o</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c17">s</font> &amp;  <a href="osalg_1.html#K1" title="OSALG_1:func.1">the_result_sort_of</a> <font color="Maroon" title="c23">o1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c17">s</font> &amp;  ex <font color="Olive" title="b1">w3</font> being   <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  st <br/>(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Olive" title="b1">w3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c27">tsb</font> &amp; ( for <font color="Olive" title="b2">y</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Olive" title="b1">w3</font> holds <br/><span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c27">tsb</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">y</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c26">tsa</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">y</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">R2</font> <a href="msualg_4.html#K1" title="MSUALG_4:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">w3</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b2">y</font></span>)</span> ) ) )
 
<b>by </b><i><a class="txt" href="osafree.html#E2:58_1"><i><font color="Green" title="E41">A3</font></i></a>, <a class="txt" href="osafree.html#E18:58_1_2"><i><font color="Green" title="E52">A28</font></i></a>, <a class="txt" href="osafree.html#E21:58_1_2"><i><font color="Green" title="E53">A29</font></i></a>, <a class="txt" href="osafree.html#E27:58_1_2"><i><font color="Green" title="E56">A33</font></i></a>, <a class="txt" href="osafree.html#E29:58_1_2"><i><font color="Green" title="E57">A34</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c28">w3</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> <b> such that </b><br/><a NAME="E35:58_1_2"/><i><font color="Green" title="E60">A37</font></i>: 
(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c28">w3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c27">tsb</font> &amp; ( for <font color="Olive" title="b1">y</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c28">w3</font> holds <br/><span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c27">tsb</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">y</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c26">tsa</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">y</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">R2</font> <a href="msualg_4.html#K1" title="MSUALG_4:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c28">w3</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b1">y</font></span>)</span> ) )
 <b>by </b><i><a class="txt" href="osafree.html#E2:58_1"><i><font color="Green" title="E41">A3</font></i></a>, <a class="txt" href="osafree.html#E18:58_1_2"><i><font color="Green" title="E52">A28</font></i></a>, <a class="txt" href="osafree.html#E21:58_1_2"><i><font color="Green" title="E53">A29</font></i></a>, <a class="txt" href="osafree.html#E27:58_1_2"><i><font color="Green" title="E56">A33</font></i></a>, <a class="txt" href="osafree.html#E29:58_1_2"><i><font color="Green" title="E57">A34</font></i></a></i>;<br/>
<a NAME="E36:58_1_2"/>
for <font color="Olive" title="b1">y</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c28">w3</font> holds <br/><span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c27">tsb</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">y</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c26">tsa</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">y</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">R1</font> <a href="msualg_4.html#K1" title="MSUALG_4:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c28">w3</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b1">y</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_2_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c29">y</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:58_1_2_4"/><i><font color="Green" title="E61">A38</font></i>: 
<font color="Maroon" title="c29">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c28">w3</font>
 ;<br/>

<a NAME="E2:58_1_2_4"/><i><font color="Green" title="E62">A39</font></i>: 
<font color="Maroon" title="c27">tsb</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c13">ts</font>
 
<b>by </b><i><a class="txt" href="osafree.html#E30:58_1_2"><i><font color="Green" title="E58">A35</font></i></a>, <a class="txt" href="osafree.html#E35:58_1_2"><i><font color="Green" title="E60">A37</font></i></a>, <a class="txt" href="osafree.html#E1:58_1_2_4"><i><font color="Green" title="E61">A38</font></i></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c30">t</font> = <font color="Maroon" title="c27">tsb</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">y</font> as    <a href="dtconstr.html#M1" title="DTCONSTR:mode.1">Element</a> of  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" title="OSAFREE:func.2">DTConOSA</a> <font color="Maroon" title="c2">X</font></span>)</span> <b>by </b><i><a class="txt" href="osafree.html#E16:58_1_2"><i><font color="Green" title="E51">A15</font></i></a></i>;<br/>
<a NAME="E4:58_1_2_4"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c30">t</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c26">tsa</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">y</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">R2</font> <a href="msualg_4.html#K1" title="MSUALG_4:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c28">w3</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c29">y</font></span>)</span>
 
<b>by </b><i><a class="txt" href="osafree.html#E35:58_1_2"><i><font color="Green" title="E60">A37</font></i></a>, <a class="txt" href="osafree.html#E1:58_1_2_4"><i><font color="Green" title="E61">A38</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:58_1_2_4"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c27">tsb</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">y</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c26">tsa</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">y</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">R1</font> <a href="msualg_4.html#K1" title="MSUALG_4:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c28">w3</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c29">y</font></span>)</span>
 <b>by </b><i><a class="txt" href="osafree.html#E2:58_1_2"><i><font color="Green" title="E45">A9</font></i></a>, <a class="txt" href="osafree.html#E2:58_1_2_4"><i><font color="Green" title="E62">A39</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E37:58_1_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">nt</font> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Maroon" title="c13">ts</font></span>)</span>,<font color="Maroon" title="c16">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">R1</font> <a href="msualg_4.html#K1" title="MSUALG_4:func.1">.</a> <font color="Maroon" title="c17">s</font>
 <b>by </b><i><a class="txt" href="osafree.html#E1:58_1"><i><font color="Green" title="E40">A2</font></i></a>, <a class="txt" href="osafree.html#E21:58_1_2"><i><font color="Green" title="E53">A29</font></i></a>, <a class="txt" href="osafree.html#E27:58_1_2"><i><font color="Green" title="E56">A33</font></i></a>, <a class="txt" href="osafree.html#E29:58_1_2"><i><font color="Green" title="E57">A34</font></i></a>, <a class="txt" href="osafree.html#E33:58_1_2"><i><font color="Green" title="E59">A36</font></i></a>, <a class="txt" href="osafree.html#E35:58_1_2"><i><font color="Green" title="E60">A37</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:58_1"/><i><font color="Green" title="E44">A40</font></i>: 
for <font color="Olive" title="b1">t</font> being  <a href="trees_2.html#NM6" title="TREES_2:NM.6">DecoratedTree</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="osafree.html#K2" title="OSAFREE:func.2">DTConOSA</a> <font color="Maroon" title="c2">X</font></span>)</span>  st <font color="Olive" title="b1">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" title="OSAFREE:func.2">DTConOSA</a> <font color="Maroon" title="c2">X</font></span>)</span> holds <br/><font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">t</font>]
 
<b>from </b><i><a class="ref" href="dtconstr.html#S7" title="DTCONSTR:sch.7">DTCONSTR:sch 7</a>(<a class="txt" href="osafree.html#E3:58_1"><i><font color="Green" title="E42">A4</font></i></a>, <a class="txt" href="osafree.html#E4:58_1"><i><font color="Green" title="E43">A7</font></i></a>);<br/></i>
<a NAME="E6:58_1"/>
for <font color="Olive" title="b1">i</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font> holds <br/><font color="Maroon" title="c10">R1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">R2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c12">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:58_1_3"/><i><font color="Green" title="E45">A41</font></i>: 
<font color="Maroon" title="c12">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c13">s</font> = <font color="Maroon" title="c12">i</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font> <b>by </b><i><a class="txt" href="osafree.html#E1:58_1_3"><i><font color="Green" title="E45">A41</font></i></a></i>;<br/>
<a NAME="E3:58_1_3"/>
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">R1</font> <a href="msualg_4.html#K1" title="MSUALG_4:func.1">.</a> <font color="Maroon" title="c13">s</font> iff <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">R2</font> <a href="msualg_4.html#K1" title="MSUALG_4:func.1">.</a> <font color="Maroon" title="c13">s</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c14">a</font>, <font color="Maroon" title="c15">b</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<i><font color="Green" title="E46">A42</font></i>: the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">s</font> = 
 <a href="osafree.html#K4" title="OSAFREE:func.4">ParsedTerms</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c13">s</font>
<b>by </b><i><a class="ref" href="osafree.html#D8" target="_self" title="OSAFREE:def.8">Def8</a></i>
<br/>.= 
<span class="p1">{<span class="default"> <font color="Olive" title="b1">a1</font> where <font color="Olive" title="b1">a1</font> is    <a href="dtconstr.html#M1" title="DTCONSTR:mode.1">Element</a> of  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" title="OSAFREE:func.2">DTConOSA</a> <font color="Maroon" title="c2">X</font></span>)</span> : (  ex <font color="Olive" title="b2">s1</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font> ex <font color="Olive" title="b3">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">s1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c13">s</font> &amp; <font color="Olive" title="b3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">s1</font> &amp; <font color="Olive" title="b1">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="trees_4.html#K1" title="TREES_4:func.1">root-tree</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">x</font>,<font color="Olive" title="b2">s1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> ) or  ex <font color="Olive" title="b2">o</font> being  <a href="osalg_1.html#NM2" title="OSALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">S</font> st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">o</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp;  <a href="osalg_1.html#K1" title="OSALG_1:func.1">the_result_sort_of</a> <font color="Olive" title="b2">o</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c13">s</font> ) ) </span>}</span> 

;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:58_1_3_1_2"/><i><font color="Green" title="E47">A43</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">a</font>,<font color="Maroon" title="c15">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">R1</font> <a href="msualg_4.html#K1" title="MSUALG_4:func.1">.</a> <font color="Maroon" title="c13">s</font>
 ;<br/><a NAME="E2:58_1_3_1_2"/><b>then </b>
( <font color="Maroon" title="c14">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">s</font> &amp; <font color="Maroon" title="c15">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">s</font> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><b>then consider </b><font color="Maroon" title="c16">a1</font> being    <a href="dtconstr.html#M1" title="DTCONSTR:mode.1">Element</a> of  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" title="OSAFREE:func.2">DTConOSA</a> <font color="Maroon" title="c2">X</font></span>)</span><b> such that </b><br/><a NAME="E4:58_1_3_1_2"/><i><font color="Green" title="E48">A44</font></i>: 
<font color="Maroon" title="c14">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">a1</font>
 <b>and </b><br/><a NAME="E5:58_1_3_1_2"/>
(  ex <font color="Olive" title="b1">s1</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font> ex <font color="Olive" title="b2">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">s1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c13">s</font> &amp; <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">s1</font> &amp; <font color="Maroon" title="c16">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="trees_4.html#K1" title="TREES_4:func.1">root-tree</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">x</font>,<font color="Olive" title="b1">s1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> ) or  ex <font color="Olive" title="b1">o</font> being  <a href="osalg_1.html#NM2" title="OSALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">S</font> st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">o</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">a1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp;  <a href="osalg_1.html#K1" title="OSALG_1:func.1">the_result_sort_of</a> <font color="Olive" title="b1">o</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c13">s</font> ) )
 <b>by </b><i><a class="txt" href="osafree.html#E1:58_1_3_1"><i><font color="Green" title="E46">A42</font></i></a></i>;<br/><b>thus </b><a NAME="E6:58_1_3_1_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">a</font>,<font color="Maroon" title="c15">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">R2</font> <a href="msualg_4.html#K1" title="MSUALG_4:func.1">.</a> <font color="Maroon" title="c13">s</font>
 <b>by </b><i><a class="txt" href="osafree.html#E5:58_1"><i><font color="Green" title="E44">A40</font></i></a>, <a class="txt" href="osafree.html#E1:58_1_3_1_2"><i><font color="Green" title="E47">A43</font></i></a>, <a class="txt" href="osafree.html#E4:58_1_3_1_2"><i><font color="Green" title="E48">A44</font></i></a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E3:58_1_3_1"/><i><font color="Green" title="E47">A45</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">a</font>,<font color="Maroon" title="c15">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">R2</font> <a href="msualg_4.html#K1" title="MSUALG_4:func.1">.</a> <font color="Maroon" title="c13">s</font>
 ;<br/>

<a NAME="E4:58_1_3_1"/><b>then </b>
( <font color="Maroon" title="c14">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">s</font> &amp; <font color="Maroon" title="c15">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">s</font> )
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c16">a1</font> being    <a href="dtconstr.html#M1" title="DTCONSTR:mode.1">Element</a> of  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" title="OSAFREE:func.2">DTConOSA</a> <font color="Maroon" title="c2">X</font></span>)</span><b> such that </b><br/><a NAME="E6:58_1_3_1"/><i><font color="Green" title="E48">A46</font></i>: 
<font color="Maroon" title="c14">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">a1</font>
 <b>and </b><br/><a NAME="E7:58_1_3_1"/>
(  ex <font color="Olive" title="b1">s1</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font> ex <font color="Olive" title="b2">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">s1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c13">s</font> &amp; <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">s1</font> &amp; <font color="Maroon" title="c16">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="trees_4.html#K1" title="TREES_4:func.1">root-tree</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">x</font>,<font color="Olive" title="b1">s1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> ) or  ex <font color="Olive" title="b1">o</font> being  <a href="osalg_1.html#NM2" title="OSALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">S</font> st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">o</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">a1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp;  <a href="osalg_1.html#K1" title="OSALG_1:func.1">the_result_sort_of</a> <font color="Olive" title="b1">o</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c13">s</font> ) )
 <b>by </b><i><a class="txt" href="osafree.html#E1:58_1_3_1"><i><font color="Green" title="E46">A42</font></i></a></i>;<br/>
<b>thus </b><a NAME="E8:58_1_3_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">a</font>,<font color="Maroon" title="c15">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">R1</font> <a href="msualg_4.html#K1" title="MSUALG_4:func.1">.</a> <font color="Maroon" title="c13">s</font>
 <b>by </b><i><a class="txt" href="osafree.html#E5:58_1"><i><font color="Green" title="E44">A40</font></i></a>, <a class="txt" href="osafree.html#E3:58_1_3_1"><i><font color="Green" title="E47">A45</font></i></a>, <a class="txt" href="osafree.html#E6:58_1_3_1"><i><font color="Green" title="E48">A46</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:58_1_3"/>
<font color="Maroon" title="c10">R1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">R2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font>
 <b>by </b><i><a class="ref" href="relat_1.html#D2" title="RELAT_1:def.2">RELAT_1:def 2</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:58_1"/>
<font color="Maroon" title="c10">R1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c11">R2</font>
 <b>by </b><i><a class="ref" href="pboole.html#T3" title="PBOOLE:th.3">PBOOLE:3</a></i>;<br/>


</div><b>end;</b></div>
<b>set </b><font color="Maroon" title="c10">P</font> =  <a href="osafree.html#K20" title="OSAFREE:func.20">PTCongruence</a> <font color="Maroon" title="c2">X</font>;<br/>
<a NAME="E2:58"/><i><font color="Green" title="E40">A47</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="osafree.html#K20" title="OSAFREE:func.20">PTCongruence</a> <font color="Maroon" title="c2">X</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">s1</font>, <font color="Maroon" title="c12">s2</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font>;<br/><b>let </b><font color="Maroon" title="c13">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>



<b>assume </b><a NAME="E1:58_2"/><i><font color="Green" title="E41">A48</font></i>: 
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">s1</font>
 ;<br/>

<a NAME="E2:58_2"/><i><font color="Green" title="E42">A49</font></i>: 
<span class="p1">(<span class="default"><a href="osafree.html#K20" title="OSAFREE:func.20">PTCongruence</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="msualg_4.html#K2" title="MSUALG_4:func.2">.</a> <font color="Maroon" title="c12">s2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x1</font>,<font color="Olive" title="b2">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">x1</font>, <font color="Olive" title="b2">y</font> is    <a href="dtconstr.html#M1" title="DTCONSTR:mode.1">Element</a> of  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" title="OSAFREE:func.2">DTConOSA</a> <font color="Maroon" title="c2">X</font></span>)</span> : <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x1</font>,<font color="Maroon" title="c12">s2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="osafree.html#K19" title="OSAFREE:func.19">PTClasses</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">y</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="osafree.html#D23" target="_self" title="OSAFREE:def.23">Def23</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c14">sy</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">x</font>,<font color="Maroon" title="c11">s1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> as   <a href="dtconstr.html#NM2" title="DTCONSTR:NM.2">Terminal</a> of <span class="p1">(<span class="default"><a href="osafree.html#K2" title="OSAFREE:func.2">DTConOSA</a> <font color="Maroon" title="c2">X</font></span>)</span> <b>by </b><i><a class="txt" href="osafree.html#E1:58_2"><i><font color="Green" title="E41">A48</font></i></a>, <a class="ref" href="osafree.html#T4" target="_self" title="OSAFREE:th.4">Th4</a></i>;<br/>
<a NAME="E4:58_2"/><i><font color="Green" title="E43">A50</font></i>: 
 <a href="trees_4.html#K1" title="TREES_4:func.1">root-tree</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">x</font>,<font color="Maroon" title="c11">s1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">s1</font>
 
<b>by </b><i><a class="txt" href="osafree.html#E1:58_2"><i><font color="Green" title="E41">A48</font></i></a>, <a class="ref" href="osafree.html#T10" target="_self" title="OSAFREE:th.10">Th10</a></i>;<br/>
<i><font color="Green" title="E44">A51</font></i>: <span class="p1">(<span class="default"><a href="osafree.html#K19" title="OSAFREE:func.19">PTClasses</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K8" title="DTCONSTR:func.8">root-tree</a> <font color="Maroon" title="c14">sy</font></span>)</span> = 
 <a href="osafree.html#K17" title="OSAFREE:func.17">@</a> <font color="Maroon" title="c14">sy</font>
<b>by </b><i><a class="ref" href="osafree.html#D22" target="_self" title="OSAFREE:def.22">Def22</a></i>
<br/>.= 
<span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="dtconstr.html#K8" title="DTCONSTR:func.8">root-tree</a> <font color="Maroon" title="c14">sy</font></span>)</span>,<font color="Olive" title="b1">s3</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">s3</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font> :  ex <font color="Olive" title="b2">s4</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font> ex <font color="Olive" title="b3">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">s4</font> &amp; <font color="Maroon" title="c14">sy</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">x</font>,<font color="Olive" title="b2">s4</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Olive" title="b2">s4</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">s3</font> ) </span>}</span> 

;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:58_2_2"/><i><font color="Green" title="E45">A52</font></i>: 
<font color="Maroon" title="c11">s1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c12">s2</font>
 ;<br/><a NAME="E2:58_2_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="dtconstr.html#K8" title="DTCONSTR:func.8">root-tree</a> <font color="Maroon" title="c14">sy</font></span>)</span>,<font color="Maroon" title="c11">s1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="osafree.html#K19" title="OSAFREE:func.19">PTClasses</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K8" title="DTCONSTR:func.8">root-tree</a> <font color="Maroon" title="c14">sy</font></span>)</span>
 <b>by </b><i><a class="txt" href="osafree.html#E4:58_2"><i><font color="Green" title="E43">A50</font></i></a>, <a class="ref" href="osafree.html#T20" target="_self" title="OSAFREE:th.20">Th20</a></i>;<br/><a NAME="E3:58_2_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="dtconstr.html#K8" title="DTCONSTR:func.8">root-tree</a> <font color="Maroon" title="c14">sy</font></span>)</span>,<font color="Maroon" title="c12">s2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="osafree.html#K19" title="OSAFREE:func.19">PTClasses</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K8" title="DTCONSTR:func.8">root-tree</a> <font color="Maroon" title="c14">sy</font></span>)</span>
 <b>by </b><i><a class="txt" href="osafree.html#E4:58_2"><i><font color="Green" title="E43">A50</font></i></a>, <a class="txt" href="osafree.html#E1:58_2_2"><i><font color="Green" title="E45">A52</font></i></a>, <a class="ref" href="osafree.html#T22" target="_self" title="OSAFREE:th.22">Th22</a></i>;<br/><b>hence </b><a NAME="E4:58_2_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="trees_4.html#K1" title="TREES_4:func.1">root-tree</a> <span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">x</font>,<font color="Maroon" title="c11">s1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span>)</span>,<span class="p2">(<span class="default"><a href="trees_4.html#K1" title="TREES_4:func.1">root-tree</a> <span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">x</font>,<font color="Maroon" title="c11">s1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="osafree.html#K20" title="OSAFREE:func.20">PTCongruence</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="msualg_4.html#K2" title="MSUALG_4:func.2">.</a> <font color="Maroon" title="c12">s2</font>
 <b>by </b><i><a class="txt" href="osafree.html#E2:58_2"><i><font color="Green" title="E42">A49</font></i></a></i>;<br/></div>
<b>end;</b></div>

<a NAME="E7:58_2"/>
 <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K20" title="OSAFREE:func.20">PTCongruence</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="msualg_4.html#K2" title="MSUALG_4:func.2">.</a> <font color="Maroon" title="c12">s2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">s2</font>
 
<b>by </b><i><a class="ref" href="orders_1.html#T97" title="ORDERS_1:th.97">ORDERS_1:97</a></i>;<br/>
<a NAME="E8:58_2"/><b>then </b><i><font color="Green" title="E45">A53</font></i>: 
<span class="p1">(<span class="default"><a href="osafree.html#K20" title="OSAFREE:func.20">PTCongruence</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="msualg_4.html#K2" title="MSUALG_4:func.2">.</a> <font color="Maroon" title="c12">s2</font> <a href="relat_2.html#R3" title="RELAT_2:pred.3">is_symmetric_in</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">s2</font>
 
<b>by </b><i><a class="ref" href="relat_2.html#D11" title="RELAT_2:def.11">RELAT_2:def 11</a></i>;<br/>
<b>let </b><font color="Maroon" title="c15">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E9:58_2"/><i><font color="Green" title="E46">A54</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="trees_4.html#K1" title="TREES_4:func.1">root-tree</a> <span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">x</font>,<font color="Maroon" title="c11">s1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span>)</span>,<font color="Maroon" title="c15">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="osafree.html#K20" title="OSAFREE:func.20">PTCongruence</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="msualg_4.html#K2" title="MSUALG_4:func.2">.</a> <font color="Maroon" title="c12">s2</font> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c15">y</font>,<span class="p2">(<span class="default"><a href="trees_4.html#K1" title="TREES_4:func.1">root-tree</a> <span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">x</font>,<font color="Maroon" title="c11">s1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="osafree.html#K20" title="OSAFREE:func.20">PTCongruence</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="msualg_4.html#K2" title="MSUALG_4:func.2">.</a> <font color="Maroon" title="c12">s2</font> )
 ;<br/>

<a NAME="E10:58_2"/><b>then </b>
( <font color="Maroon" title="c15">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">s2</font> &amp;  <a href="trees_4.html#K1" title="TREES_4:func.1">root-tree</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">x</font>,<font color="Maroon" title="c11">s1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">s2</font> )
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E11:58_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c15">y</font>,<span class="p2">(<span class="default"><a href="dtconstr.html#K8" title="DTCONSTR:func.8">root-tree</a> <font color="Maroon" title="c14">sy</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="osafree.html#K20" title="OSAFREE:func.20">PTCongruence</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="msualg_4.html#K2" title="MSUALG_4:func.2">.</a> <font color="Maroon" title="c12">s2</font>
 
<b>by </b><i><a class="txt" href="osafree.html#E8:58_2"><i><font color="Green" title="E45">A53</font></i></a>, <a class="txt" href="osafree.html#E9:58_2"><i><font color="Green" title="E46">A54</font></i></a>, <a class="ref" href="relat_2.html#D3" title="RELAT_2:def.3">RELAT_2:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c16">y1</font>, <font color="Maroon" title="c17">r1</font> being    <a href="dtconstr.html#M1" title="DTCONSTR:mode.1">Element</a> of  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" title="OSAFREE:func.2">DTConOSA</a> <font color="Maroon" title="c2">X</font></span>)</span><b> such that </b><br/><a NAME="E13:58_2"/><i><font color="Green" title="E47">A55</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c15">y</font>,<span class="p2">(<span class="default"><a href="dtconstr.html#K8" title="DTCONSTR:func.8">root-tree</a> <font color="Maroon" title="c14">sy</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">y1</font>,<font color="Maroon" title="c17">r1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>and </b><br/><a NAME="E14:58_2"/><i><font color="Green" title="E48">A56</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">y1</font>,<font color="Maroon" title="c12">s2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="osafree.html#K19" title="OSAFREE:func.19">PTClasses</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">r1</font>
 <b>by </b><i><a class="txt" href="osafree.html#E2:58_2"><i><font color="Green" title="E42">A49</font></i></a></i>;<br/>
<a NAME="E15:58_2"/><i><font color="Green" title="E49">A57</font></i>: 
( <font color="Maroon" title="c15">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">y1</font> &amp;  <a href="dtconstr.html#K8" title="DTCONSTR:func.8">root-tree</a> <font color="Maroon" title="c14">sy</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">r1</font> )
 
<b>by </b><i><a class="txt" href="osafree.html#E13:58_2"><i><font color="Green" title="E47">A55</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c18">s3</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font><b> such that </b><br/><a NAME="E17:58_2"/><i><font color="Green" title="E50">A58</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">y1</font>,<font color="Maroon" title="c12">s2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="dtconstr.html#K8" title="DTCONSTR:func.8">root-tree</a> <font color="Maroon" title="c14">sy</font></span>)</span>,<font color="Maroon" title="c18">s3</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>and </b><br/><a NAME="E18:58_2"/><i><font color="Green" title="E51">A59</font></i>: 
 ex <font color="Olive" title="b1">s4</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font> ex <font color="Olive" title="b2">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">s4</font> &amp; <font color="Maroon" title="c14">sy</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">x</font>,<font color="Olive" title="b1">s4</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Olive" title="b1">s4</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c18">s3</font> )
 <b>by </b><i><a class="txt" href="osafree.html#E5:58_2"><i><font color="Green" title="E44">A51</font></i></a>, <a class="txt" href="osafree.html#E14:58_2"><i><font color="Green" title="E48">A56</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c19">s4</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font>, <font color="Maroon" title="c20">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E20:58_2"/><i><font color="Green" title="E52">A60</font></i>: 
( <font color="Maroon" title="c20">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">s4</font> &amp; <font color="Maroon" title="c14">sy</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c20">x</font>,<font color="Maroon" title="c19">s4</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c19">s4</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c18">s3</font> )
 <b>by </b><i><a class="txt" href="osafree.html#E18:58_2"><i><font color="Green" title="E51">A59</font></i></a></i>;<br/>
<a NAME="E21:58_2"/>
( <font color="Maroon" title="c15">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="dtconstr.html#K8" title="DTCONSTR:func.8">root-tree</a> <font color="Maroon" title="c14">sy</font> &amp; <font color="Maroon" title="c12">s2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">s3</font> )
 
<b>by </b><i><a class="txt" href="osafree.html#E15:58_2"><i><font color="Green" title="E49">A57</font></i></a>, <a class="txt" href="osafree.html#E17:58_2"><i><font color="Green" title="E50">A58</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/>
<b>hence </b><a NAME="E22:58_2"/>
( <font color="Maroon" title="c11">s1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c12">s2</font> &amp; <font color="Maroon" title="c15">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="trees_4.html#K1" title="TREES_4:func.1">root-tree</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">x</font>,<font color="Maroon" title="c11">s1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <b>by </b><i><a class="txt" href="osafree.html#E20:58_2"><i><font color="Green" title="E52">A60</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:58"/>
<font color="Maroon">S<sub>2</sub></font>[ <a href="osafree.html#K20" title="OSAFREE:func.20">PTCongruence</a> <font color="Maroon" title="c2">X</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">o1</font>, <font color="Maroon" title="c12">o2</font> be   <a href="osalg_1.html#NM2" title="OSALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">S</font>;<br/><b>let </b><font color="Maroon" title="c13">x1</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Maroon" title="c11">o1</font>,<span class="p1">(<span class="default"><a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</a> <font color="Maroon" title="c2">X</font></span>)</span>;<br/><b>let </b><font color="Maroon" title="c14">x2</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Maroon" title="c12">o2</font>,<span class="p1">(<span class="default"><a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</a> <font color="Maroon" title="c2">X</font></span>)</span>;<br/><b>let </b><font color="Maroon" title="c15">s3</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font>;<br/>







<a NAME="E1:58_3"/><i><font color="Green" title="E41">A61</font></i>: 
<span class="p1">(<span class="default"><a href="osafree.html#K20" title="OSAFREE:func.20">PTCongruence</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="msualg_4.html#K2" title="MSUALG_4:func.2">.</a> <font color="Maroon" title="c15">s3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x3</font>,<font color="Olive" title="b2">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">x3</font>, <font color="Olive" title="b2">y</font> is    <a href="dtconstr.html#M1" title="DTCONSTR:mode.1">Element</a> of  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" title="OSAFREE:func.2">DTConOSA</a> <font color="Maroon" title="c2">X</font></span>)</span> : <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x3</font>,<font color="Maroon" title="c15">s3</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="osafree.html#K19" title="OSAFREE:func.19">PTClasses</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">y</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="osafree.html#D23" target="_self" title="OSAFREE:def.23">Def23</a></i>;<br/>
<a NAME="E2:58_3"/><i><font color="Green" title="E42">A62</font></i>: 
( <font color="Maroon" title="c13">x1</font> is    <a href="trees_4.html#M1" title="TREES_4:mode.1">FinSequence</a> of  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" title="OSAFREE:func.2">DTConOSA</a> <font color="Maroon" title="c2">X</font></span>)</span> &amp;  <a href="osafree.html#K9" title="OSAFREE:func.9">OSSym</a> <font color="Maroon" title="c11">o1</font>,<font color="Maroon" title="c2">X</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a>  <a href="trees_3.html#K16" title="TREES_3:func.16">roots</a> <font color="Maroon" title="c13">x1</font> )
 
<b>by </b><i><a class="ref" href="osafree.html#T13" target="_self" title="OSAFREE:th.13">Th13</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c16">ts1</font> = <font color="Maroon" title="c13">x1</font> as    <a href="trees_4.html#M1" title="TREES_4:mode.1">FinSequence</a> of  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" title="OSAFREE:func.2">DTConOSA</a> <font color="Maroon" title="c2">X</font></span>)</span> <b>by </b><i><a class="ref" href="osafree.html#T13" target="_self" title="OSAFREE:th.13">Th13</a></i>;<br/>
<i><font color="Green" title="E43">A63</font></i>: <span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c11">o1</font>,<span class="p2">(<span class="default"><a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">x1</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K7" title="OSAFREE:func.7">PTOper</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="msualg_4.html#K5" title="MSUALG_4:func.5">.</a> <font color="Maroon" title="c11">o1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x1</font>
<b>by </b><i><a class="ref" href="msualg_1.html#D11" title="MSUALG_1:def.11">MSUALG_1:def 11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="osafree.html#K6" title="OSAFREE:func.6">PTDenOp</a> <font color="Maroon" title="c11">o1</font>,<font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">ts1</font>
<b>by </b><i><a class="ref" href="osafree.html#D10" target="_self" title="OSAFREE:def.10">Def10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="osafree.html#K9" title="OSAFREE:func.9">OSSym</a> <font color="Maroon" title="c11">o1</font>,<font color="Maroon" title="c2">X</font></span>)</span> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Maroon" title="c16">ts1</font>
<b>by </b><i><a class="txt" href="osafree.html#E2:58_3"><i><font color="Green" title="E42">A62</font></i></a>, <a class="ref" href="osafree.html#D9" target="_self" title="OSAFREE:def.9">Def9</a></i>
;<br/>
<a NAME="E5:58_3"/><i><font color="Green" title="E44">A64</font></i>: 
( <font color="Maroon" title="c14">x2</font> is    <a href="trees_4.html#M1" title="TREES_4:mode.1">FinSequence</a> of  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" title="OSAFREE:func.2">DTConOSA</a> <font color="Maroon" title="c2">X</font></span>)</span> &amp;  <a href="osafree.html#K9" title="OSAFREE:func.9">OSSym</a> <font color="Maroon" title="c12">o2</font>,<font color="Maroon" title="c2">X</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a>  <a href="trees_3.html#K16" title="TREES_3:func.16">roots</a> <font color="Maroon" title="c14">x2</font> )
 
<b>by </b><i><a class="ref" href="osafree.html#T13" target="_self" title="OSAFREE:th.13">Th13</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c17">ts2</font> = <font color="Maroon" title="c14">x2</font> as    <a href="trees_4.html#M1" title="TREES_4:mode.1">FinSequence</a> of  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" title="OSAFREE:func.2">DTConOSA</a> <font color="Maroon" title="c2">X</font></span>)</span> <b>by </b><i><a class="ref" href="osafree.html#T13" target="_self" title="OSAFREE:th.13">Th13</a></i>;<br/>
<i><font color="Green" title="E45">A65</font></i>: <span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c12">o2</font>,<span class="p2">(<span class="default"><a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">x2</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K7" title="OSAFREE:func.7">PTOper</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="msualg_4.html#K5" title="MSUALG_4:func.5">.</a> <font color="Maroon" title="c12">o2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x2</font>
<b>by </b><i><a class="ref" href="msualg_1.html#D11" title="MSUALG_1:def.11">MSUALG_1:def 11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="osafree.html#K6" title="OSAFREE:func.6">PTDenOp</a> <font color="Maroon" title="c12">o2</font>,<font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">ts2</font>
<b>by </b><i><a class="ref" href="osafree.html#D10" target="_self" title="OSAFREE:def.10">Def10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="osafree.html#K9" title="OSAFREE:func.9">OSSym</a> <font color="Maroon" title="c12">o2</font>,<font color="Maroon" title="c2">X</font></span>)</span> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Maroon" title="c17">ts2</font>
<b>by </b><i><a class="txt" href="osafree.html#E5:58_3"><i><font color="Green" title="E44">A64</font></i></a>, <a class="ref" href="osafree.html#D9" target="_self" title="OSAFREE:def.9">Def9</a></i>
;<br/>
<b>reconsider </b><font color="Maroon" title="c18">x</font> = <span class="p1">(<span class="default"><a href="osafree.html#K19" title="OSAFREE:func.19">PTClasses</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K15" title="LANG1:func.15">*</a> <font color="Maroon" title="c16">ts1</font>, <font color="Maroon" title="c19">y</font> = <span class="p1">(<span class="default"><a href="osafree.html#K19" title="OSAFREE:func.19">PTClasses</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="lang1.html#K15" title="LANG1:func.15">*</a> <font color="Maroon" title="c17">ts2</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <span class="p3">(<span class="default"><a href="osafree.html#K2" title="OSAFREE:func.2">DTConOSA</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> ;<br/>
<a NAME="E9:58_3"/><i><font color="Green" title="E46">A66</font></i>: 
(  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c16">ts1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" title="OSAFREE:func.2">DTConOSA</a> <font color="Maroon" title="c2">X</font></span>)</span> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c17">ts2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" title="OSAFREE:func.2">DTConOSA</a> <font color="Maroon" title="c2">X</font></span>)</span> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="osafree.html#K19" title="OSAFREE:func.19">PTClasses</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" title="OSAFREE:func.2">DTConOSA</a> <font color="Maroon" title="c2">X</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E10:58_3"/><b>then </b><i><font color="Green" title="E47">A67</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c18">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">ts1</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c19">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">ts2</font> )
 
<b>by </b><i><a class="ref" href="finseq_2.html#T33" title="FINSEQ_2:th.33">FINSEQ_2:33</a></i>;<br/>
<a NAME="E11:58_3"/><b>then </b><i><font color="Green" title="E48">A68</font></i>: 
(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c18">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">ts1</font> &amp;  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c19">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c17">ts2</font> )
 
<b>by </b><i><a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/>
<i><font color="Green" title="E49">A69</font></i>: <span class="p1">(<span class="default"><a href="osafree.html#K19" title="OSAFREE:func.19">PTClasses</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K9" title="OSAFREE:func.9">OSSym</a> <font color="Maroon" title="c11">o1</font>,<font color="Maroon" title="c2">X</font></span>)</span> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Maroon" title="c16">ts1</font></span>)</span> = 
 <a href="osafree.html#K18" title="OSAFREE:func.18">@</a> <span class="p1">(<span class="default"><a href="osafree.html#K9" title="OSAFREE:func.9">OSSym</a> <font color="Maroon" title="c11">o1</font>,<font color="Maroon" title="c2">X</font></span>)</span>,<font color="Maroon" title="c18">x</font>
<b>by </b><i><a class="txt" href="osafree.html#E2:58_3"><i><font color="Green" title="E42">A62</font></i></a>, <a class="ref" href="osafree.html#D22" target="_self" title="OSAFREE:def.22">Def22</a></i>
<br/>.= 
<span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Olive" title="b1">o3</font>,<span class="p5">(<span class="default"><a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">x3</font></span>)</span>,<font color="Olive" title="b3">s4</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">o3</font> is   <a href="osalg_1.html#NM2" title="OSALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">S</font>, <font color="Olive" title="b2">x3</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Olive" title="b1">o3</font>,<span class="p1">(<span class="default"><a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</a> <font color="Maroon" title="c2">X</font></span>)</span>, <font color="Olive" title="b3">s4</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font> : (  ex <font color="Olive" title="b4">o4</font> being  <a href="osalg_1.html#NM2" title="OSALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">S</font> st <br/>(  <a href="osafree.html#K9" title="OSAFREE:func.9">OSSym</a> <font color="Maroon" title="c11">o1</font>,<font color="Maroon" title="c2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b4">o4</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Olive" title="b4">o4</font> <a href="osalg_1.html#R1" title="OSALG_1:pred.1">~=</a> <font color="Olive" title="b1">o3</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Olive" title="b4">o4</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Olive" title="b1">o3</font></span>)</span> &amp;  <a href="osalg_1.html#K1" title="OSALG_1:func.1">the_result_sort_of</a> <font color="Olive" title="b4">o4</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b3">s4</font> &amp;  <a href="osalg_1.html#K1" title="OSALG_1:func.1">the_result_sort_of</a> <font color="Olive" title="b1">o3</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b3">s4</font> ) &amp;  ex <font color="Olive" title="b4">w3</font> being   <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  st <br/>(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Olive" title="b4">w3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c18">x</font> &amp; ( for <font color="Olive" title="b5">y</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c18">x</font> holds <br/><span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b2">x3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">y</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b4">w3</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b5">y</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">y</font> ) ) ) </span>}</span> 

;<br/>
<a NAME="E13:58_3"/><i><font color="Green" title="E50">A70</font></i>: 
(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c11">o1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c13">x1</font> &amp;  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c12">o2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c14">x2</font> )
 
<b>by </b><i><a class="ref" href="msualg_3.html#T6" title="MSUALG_3:th.6">MSUALG_3:6</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c20">tx</font> = <span class="p1">(<span class="default"><a href="osafree.html#K9" title="OSAFREE:func.9">OSSym</a> <font color="Maroon" title="c11">o1</font>,<font color="Maroon" title="c2">X</font></span>)</span> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Maroon" title="c16">ts1</font>, <font color="Maroon" title="c21">ty</font> = <span class="p1">(<span class="default"><a href="osafree.html#K9" title="OSAFREE:func.9">OSSym</a> <font color="Maroon" title="c12">o2</font>,<font color="Maroon" title="c2">X</font></span>)</span> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Maroon" title="c17">ts2</font> as    <a href="dtconstr.html#M1" title="DTCONSTR:mode.1">Element</a> of  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" title="OSAFREE:func.2">DTConOSA</a> <font color="Maroon" title="c2">X</font></span>)</span> <b>by </b><i><a class="txt" href="osafree.html#E2:58_3"><i><font color="Green" title="E42">A62</font></i></a>, <a class="txt" href="osafree.html#E5:58_3"><i><font color="Green" title="E44">A64</font></i></a>, <a class="ref" href="osafree.html#T12" target="_self" title="OSAFREE:th.12">Th12</a></i>;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:58_3_4"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c11">o1</font>,<span class="p4">(<span class="default"><a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">x1</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c12">o2</font>,<span class="p4">(<span class="default"><a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">x2</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="osafree.html#K20" title="OSAFREE:func.20">PTCongruence</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="msualg_4.html#K2" title="MSUALG_4:func.2">.</a> <font color="Maroon" title="c15">s3</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c22">t1</font>, <font color="Maroon" title="c23">t2</font> being    <a href="dtconstr.html#M1" title="DTCONSTR:mode.1">Element</a> of  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" title="OSAFREE:func.2">DTConOSA</a> <font color="Maroon" title="c2">X</font></span>)</span><b> such that </b><br/><a NAME="E3:58_3_4"/><i><font color="Green" title="E51">A71</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c11">o1</font>,<span class="p4">(<span class="default"><a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">x1</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c12">o2</font>,<span class="p4">(<span class="default"><a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">x2</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c22">t1</font>,<font color="Maroon" title="c23">t2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>and </b><br/><a NAME="E4:58_3_4"/><i><font color="Green" title="E52">A72</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c22">t1</font>,<font color="Maroon" title="c15">s3</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="osafree.html#K19" title="OSAFREE:func.19">PTClasses</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c23">t2</font>
 <b>by </b><i><a class="txt" href="osafree.html#E1:58_3"><i><font color="Green" title="E41">A61</font></i></a></i>;<br/><a NAME="E5:58_3_4"/><i><font color="Green" title="E53">A73</font></i>: 
( <span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c11">o1</font>,<span class="p2">(<span class="default"><a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">t1</font> &amp; <span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c12">o2</font>,<span class="p2">(<span class="default"><a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">t2</font> )
 <b>by </b><i><a class="txt" href="osafree.html#E3:58_3_4"><i><font color="Green" title="E51">A71</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/><a NAME="E6:58_3_4"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c23">t2</font>,<font color="Maroon" title="c15">s3</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="osafree.html#K19" title="OSAFREE:func.19">PTClasses</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">t1</font>
 <b>by </b><i><a class="txt" href="osafree.html#E4:58_3_4"><i><font color="Green" title="E52">A72</font></i></a>, <a class="ref" href="osafree.html#T20" target="_self" title="OSAFREE:th.20">Th20</a></i>;<br/><b>then consider </b><font color="Maroon" title="c24">o3</font> being   <a href="osalg_1.html#NM2" title="OSALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">S</font>, <font color="Maroon" title="c25">x3</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Maroon" title="c24">o3</font>,<span class="p1">(<span class="default"><a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</a> <font color="Maroon" title="c2">X</font></span>)</span>, <font color="Maroon" title="c26">s4</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font><b> such that </b><br/><a NAME="E8:58_3_4"/><i><font color="Green" title="E54">A74</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c23">t2</font>,<font color="Maroon" title="c15">s3</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c24">o3</font>,<span class="p4">(<span class="default"><a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c25">x3</font></span>)</span>,<font color="Maroon" title="c26">s4</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>and </b><br/><a NAME="E9:58_3_4"/><i><font color="Green" title="E55">A75</font></i>: 
 ex <font color="Olive" title="b1">o4</font> being  <a href="osalg_1.html#NM2" title="OSALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">S</font> st <br/>(  <a href="osafree.html#K9" title="OSAFREE:func.9">OSSym</a> <font color="Maroon" title="c11">o1</font>,<font color="Maroon" title="c2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">o4</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Olive" title="b1">o4</font> <a href="osalg_1.html#R1" title="OSALG_1:pred.1">~=</a> <font color="Maroon" title="c24">o3</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Olive" title="b1">o4</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c24">o3</font></span>)</span> &amp;  <a href="osalg_1.html#K1" title="OSALG_1:func.1">the_result_sort_of</a> <font color="Olive" title="b1">o4</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c26">s4</font> &amp;  <a href="osalg_1.html#K1" title="OSALG_1:func.1">the_result_sort_of</a> <font color="Maroon" title="c24">o3</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c26">s4</font> )
 <b>and </b><br/><a NAME="E10:58_3_4"/><i><font color="Green" title="E56">A76</font></i>: 
 ex <font color="Olive" title="b1">w3</font> being   <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  st <br/>(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Olive" title="b1">w3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c18">x</font> &amp; ( for <font color="Olive" title="b2">y</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c18">x</font> holds <br/><span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c25">x3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">y</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b1">w3</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b2">y</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">y</font> ) )
 <b>by </b><i><a class="txt" href="osafree.html#E4:58_3"><i><font color="Green" title="E43">A63</font></i></a>, <a class="txt" href="osafree.html#E12:58_3"><i><font color="Green" title="E49">A69</font></i></a>, <a class="txt" href="osafree.html#E5:58_3_4"><i><font color="Green" title="E53">A73</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c27">o4</font> being   <a href="osalg_1.html#NM2" title="OSALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">S</font><b> such that </b><br/><a NAME="E12:58_3_4"/><i><font color="Green" title="E57">A77</font></i>: 
(  <a href="osafree.html#K9" title="OSAFREE:func.9">OSSym</a> <font color="Maroon" title="c11">o1</font>,<font color="Maroon" title="c2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c27">o4</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c27">o4</font> <a href="osalg_1.html#R1" title="OSALG_1:pred.1">~=</a> <font color="Maroon" title="c24">o3</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c27">o4</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c24">o3</font></span>)</span> &amp;  <a href="osalg_1.html#K1" title="OSALG_1:func.1">the_result_sort_of</a> <font color="Maroon" title="c27">o4</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c26">s4</font> &amp;  <a href="osalg_1.html#K1" title="OSALG_1:func.1">the_result_sort_of</a> <font color="Maroon" title="c24">o3</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c26">s4</font> )
 <b>by </b><i><a class="txt" href="osafree.html#E9:58_3_4"><i><font color="Green" title="E55">A75</font></i></a></i>;<br/><a NAME="E13:58_3_4"/><i><font color="Green" title="E58">A78</font></i>: 
( <font color="Maroon" title="c11">o1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">o4</font> &amp; <font color="Maroon" title="c23">t2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c24">o3</font>,<span class="p2">(<span class="default"><a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c25">x3</font> &amp; <font color="Maroon" title="c15">s3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">s4</font> )
 <b>by </b><i><a class="txt" href="osafree.html#E8:58_3_4"><i><font color="Green" title="E54">A74</font></i></a>, <a class="txt" href="osafree.html#E12:58_3_4"><i><font color="Green" title="E57">A77</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/><a NAME="E14:58_3_4"/><i><font color="Green" title="E59">A79</font></i>: 
( <font color="Maroon" title="c25">x3</font> is    <a href="trees_4.html#M1" title="TREES_4:mode.1">FinSequence</a> of  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" title="OSAFREE:func.2">DTConOSA</a> <font color="Maroon" title="c2">X</font></span>)</span> &amp;  <a href="osafree.html#K9" title="OSAFREE:func.9">OSSym</a> <font color="Maroon" title="c24">o3</font>,<font color="Maroon" title="c2">X</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a>  <a href="trees_3.html#K16" title="TREES_3:func.16">roots</a> <font color="Maroon" title="c25">x3</font> )
 <b>by </b><i><a class="ref" href="osafree.html#T13" target="_self" title="OSAFREE:th.13">Th13</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c28">ts3</font> = <font color="Maroon" title="c25">x3</font> as    <a href="trees_4.html#M1" title="TREES_4:mode.1">FinSequence</a> of  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" title="OSAFREE:func.2">DTConOSA</a> <font color="Maroon" title="c2">X</font></span>)</span> <b>by </b><i><a class="ref" href="osafree.html#T13" target="_self" title="OSAFREE:th.13">Th13</a></i>;<br/><span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c24">o3</font>,<span class="p2">(<span class="default"><a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c25">x3</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K7" title="OSAFREE:func.7">PTOper</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="msualg_4.html#K5" title="MSUALG_4:func.5">.</a> <font color="Maroon" title="c24">o3</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">x3</font>
<b>by </b><i><a class="ref" href="msualg_1.html#D11" title="MSUALG_1:def.11">MSUALG_1:def 11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="osafree.html#K6" title="OSAFREE:func.6">PTDenOp</a> <font color="Maroon" title="c24">o3</font>,<font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">ts3</font>
<b>by </b><i><a class="ref" href="osafree.html#D10" target="_self" title="OSAFREE:def.10">Def10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="osafree.html#K9" title="OSAFREE:func.9">OSSym</a> <font color="Maroon" title="c24">o3</font>,<font color="Maroon" title="c2">X</font></span>)</span> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Maroon" title="c28">ts3</font>
<b>by </b><i><a class="txt" href="osafree.html#E14:58_3_4"><i><font color="Green" title="E59">A79</font></i></a>, <a class="ref" href="osafree.html#D9" target="_self" title="OSAFREE:def.9">Def9</a></i>
;<br/><a NAME="E17:58_3_4"/><b>then </b><i><font color="Green" title="E60">A80</font></i>: 
(  <a href="osafree.html#K9" title="OSAFREE:func.9">OSSym</a> <font color="Maroon" title="c24">o3</font>,<font color="Maroon" title="c2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="osafree.html#K9" title="OSAFREE:func.9">OSSym</a> <font color="Maroon" title="c12">o2</font>,<font color="Maroon" title="c2">X</font> &amp; <font color="Maroon" title="c28">ts3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">ts2</font> )
 <b>by </b><i><a class="txt" href="osafree.html#E7:58_3"><i><font color="Green" title="E45">A65</font></i></a>, <a class="txt" href="osafree.html#E5:58_3_4"><i><font color="Green" title="E53">A73</font></i></a>, <a class="txt" href="osafree.html#E13:58_3_4"><i><font color="Green" title="E58">A78</font></i></a>, <a class="ref" href="trees_4.html#T15" title="TREES_4:th.15">TREES_4:15</a></i>;<br/><a NAME="E18:58_3_4"/><b>then </b><i><font color="Green" title="E61">A81</font></i>: 
( <font color="Maroon" title="c24">o3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">o2</font> &amp; <font color="Maroon" title="c25">x3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">x2</font> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/><b>thus </b><a NAME="E19:58_3_4"/>
( <font color="Maroon" title="c11">o1</font> <a href="osalg_1.html#R1" title="OSALG_1:pred.1">~=</a> <font color="Maroon" title="c12">o2</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c11">o1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c12">o2</font></span>)</span> &amp;  <a href="osalg_1.html#K1" title="OSALG_1:func.1">the_result_sort_of</a> <font color="Maroon" title="c11">o1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c15">s3</font> &amp;  <a href="osalg_1.html#K1" title="OSALG_1:func.1">the_result_sort_of</a> <font color="Maroon" title="c12">o2</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c15">s3</font> )
 <b>by </b><i><a class="txt" href="osafree.html#E12:58_3_4"><i><font color="Green" title="E57">A77</font></i></a>, <a class="txt" href="osafree.html#E13:58_3_4"><i><font color="Green" title="E58">A78</font></i></a>, <a class="txt" href="osafree.html#E17:58_3_4"><i><font color="Green" title="E60">A80</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/><a NAME="E20:58_3_4"/><i><font color="Green" title="E62">A82</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c11">o1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c12">o2</font></span>)</span>
 <b>by </b><i><a class="txt" href="osafree.html#E12:58_3_4"><i><font color="Green" title="E57">A77</font></i></a>, <a class="txt" href="osafree.html#E13:58_3_4"><i><font color="Green" title="E58">A78</font></i></a>, <a class="txt" href="osafree.html#E18:58_3_4"><i><font color="Green" title="E61">A81</font></i></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/><b>consider </b><font color="Maroon" title="c29">w3</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> <b> such that </b><br/><a NAME="E22:58_3_4"/><i><font color="Green" title="E63">A83</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c29">w3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c18">x</font>
 <b>and </b><br/><a NAME="E23:58_3_4"/><i><font color="Green" title="E64">A84</font></i>: 
for <font color="Olive" title="b1">y</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c18">x</font> holds <br/><span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c25">x3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">y</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c29">w3</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b1">y</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">y</font>
 <b>by </b><i><a class="txt" href="osafree.html#E10:58_3_4"><i><font color="Green" title="E56">A76</font></i></a></i>;<br/><b>take </b><font color="Maroon" title="c30">w3</font> = <font color="Maroon" title="c29">w3</font>;<br/><b>thus </b><a NAME="E24:58_3_4"/>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c30">w3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c13">x1</font>
 <b>by </b><i><a class="txt" href="osafree.html#E10:58_3"><i><font color="Green" title="E47">A67</font></i></a>, <a class="txt" href="osafree.html#E22:58_3_4"><i><font color="Green" title="E63">A83</font></i></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/><b>let </b><font color="Maroon" title="c31">y</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>assume </b><a NAME="E25:58_3_4"/><i><font color="Green" title="E65">A85</font></i>: 
<font color="Maroon" title="c31">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c30">w3</font>
 ;<br/><a NAME="E26:58_3_4"/><i><font color="Green" title="E66">A86</font></i>: 
<span class="p1">(<span class="default"><a href="osafree.html#K20" title="OSAFREE:func.20">PTCongruence</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="msualg_4.html#K2" title="MSUALG_4:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c30">w3</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c31">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x5</font>,<font color="Olive" title="b2">y5</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">x5</font>, <font color="Olive" title="b2">y5</font> is    <a href="dtconstr.html#M1" title="DTCONSTR:mode.1">Element</a> of  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" title="OSAFREE:func.2">DTConOSA</a> <font color="Maroon" title="c2">X</font></span>)</span> : <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x5</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c30">w3</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c31">y</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="osafree.html#K19" title="OSAFREE:func.19">PTClasses</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">y5</font> </span>}</span> 
 <b>by </b><i><a class="ref" href="osafree.html#D23" target="_self" title="OSAFREE:def.23">Def23</a></i>;<br/><a NAME="E27:58_3_4"/>
( <font color="Maroon" title="c16">ts1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c16">ts1</font> &amp; <font color="Maroon" title="c17">ts2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c17">ts2</font> )
 <b>by </b><i><a class="txt" href="osafree.html#E11:58_3"><i><font color="Green" title="E48">A68</font></i></a>, <a class="txt" href="osafree.html#E13:58_3"><i><font color="Green" title="E50">A70</font></i></a>, <a class="txt" href="osafree.html#E20:58_3_4"><i><font color="Green" title="E62">A82</font></i></a>, <a class="txt" href="osafree.html#E22:58_3_4"><i><font color="Green" title="E63">A83</font></i></a>, <a class="txt" href="osafree.html#E25:58_3_4"><i><font color="Green" title="E65">A85</font></i></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c32">t22</font> = <font color="Maroon" title="c17">ts2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">y</font>, <font color="Maroon" title="c33">t11</font> = <font color="Maroon" title="c16">ts1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">y</font> as    <a href="dtconstr.html#M1" title="DTCONSTR:mode.1">Element</a> of  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" title="OSAFREE:func.2">DTConOSA</a> <font color="Maroon" title="c2">X</font></span>)</span> <b>by </b><i><a class="txt" href="osafree.html#E9:58_3"><i><font color="Green" title="E46">A66</font></i></a></i>;<br/><a NAME="E29:58_3_4"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c25">x3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">y</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c30">w3</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c31">y</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">y</font>
 <b>by </b><i><a class="txt" href="osafree.html#E22:58_3_4"><i><font color="Green" title="E63">A83</font></i></a>, <a class="txt" href="osafree.html#E23:58_3_4"><i><font color="Green" title="E64">A84</font></i></a>, <a class="txt" href="osafree.html#E25:58_3_4"><i><font color="Green" title="E65">A85</font></i></a></i>;<br/><a NAME="E30:58_3_4"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c17">ts2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">y</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c30">w3</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c31">y</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="osafree.html#K19" title="OSAFREE:func.19">PTClasses</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">ts1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">y</font></span>)</span>
 <b>by </b><i><a class="txt" href="osafree.html#E17:58_3_4"><i><font color="Green" title="E60">A80</font></i></a>, <a class="txt" href="osafree.html#E22:58_3_4"><i><font color="Green" title="E63">A83</font></i></a>, <a class="txt" href="osafree.html#E25:58_3_4"><i><font color="Green" title="E65">A85</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>;<br/><a NAME="E31:58_3_4"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c33">t11</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c30">w3</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c31">y</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="osafree.html#K19" title="OSAFREE:func.19">PTClasses</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c32">t22</font>
 <b>by </b><i><a class="ref" href="osafree.html#T20" target="_self" title="OSAFREE:th.20">Th20</a></i>;<br/><b>hence </b><a NAME="E32:58_3_4"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">x1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">y</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">x2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">y</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="osafree.html#K20" title="OSAFREE:func.20">PTCongruence</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="msualg_4.html#K2" title="MSUALG_4:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c30">w3</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c31">y</font></span>)</span>
 <b>by </b><i><a class="txt" href="osafree.html#E26:58_3_4"><i><font color="Green" title="E66">A86</font></i></a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E16:58_3"/><i><font color="Green" title="E51">A87</font></i>: 
( <font color="Maroon" title="c11">o1</font> <a href="osalg_1.html#R1" title="OSALG_1:pred.1">~=</a> <font color="Maroon" title="c12">o2</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c11">o1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c12">o2</font></span>)</span> &amp;  <a href="osalg_1.html#K1" title="OSALG_1:func.1">the_result_sort_of</a> <font color="Maroon" title="c11">o1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c15">s3</font> &amp;  <a href="osalg_1.html#K1" title="OSALG_1:func.1">the_result_sort_of</a> <font color="Maroon" title="c12">o2</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c15">s3</font> &amp;  ex <font color="Olive" title="b1">w3</font> being   <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  st <br/>(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Olive" title="b1">w3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c13">x1</font> &amp; ( for <font color="Olive" title="b2">y</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Olive" title="b1">w3</font> holds <br/><span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">x1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">y</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">x2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">y</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="osafree.html#K20" title="OSAFREE:func.20">PTCongruence</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="msualg_4.html#K2" title="MSUALG_4:func.2">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">w3</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b2">y</font></span>)</span> ) ) )
 ;<br/>

<b>then consider </b><font color="Maroon" title="c22">w3</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> <b> such that </b><br/><a NAME="E18:58_3"/><i><font color="Green" title="E52">A88</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c22">w3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c13">x1</font>
 <b>and </b><br/><a NAME="E19:58_3"/><i><font color="Green" title="E53">A89</font></i>: 
for <font color="Olive" title="b1">y</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c22">w3</font> holds <br/><span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">x1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">y</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">x2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">y</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="osafree.html#K20" title="OSAFREE:func.20">PTCongruence</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="msualg_4.html#K2" title="MSUALG_4:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">w3</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b1">y</font></span>)</span>
 ;<br/>
<a NAME="E20:58_3"/>
for <font color="Olive" title="b1">y</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c18">x</font> holds <br/><span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">x2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">y</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">w3</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b1">y</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_3_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c23">y</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:58_3_5"/><i><font color="Green" title="E54">A90</font></i>: 
<font color="Maroon" title="c23">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c18">x</font>
 ;<br/>

<a NAME="E2:58_3_5"/><i><font color="Green" title="E55">A91</font></i>: 
<span class="p1">(<span class="default"><a href="osafree.html#K20" title="OSAFREE:func.20">PTCongruence</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="msualg_4.html#K2" title="MSUALG_4:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">w3</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c23">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x5</font>,<font color="Olive" title="b2">y5</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">x5</font>, <font color="Olive" title="b2">y5</font> is    <a href="dtconstr.html#M1" title="DTCONSTR:mode.1">Element</a> of  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" title="OSAFREE:func.2">DTConOSA</a> <font color="Maroon" title="c2">X</font></span>)</span> : <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x5</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">w3</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c23">y</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="osafree.html#K19" title="OSAFREE:func.19">PTClasses</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">y5</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="osafree.html#D23" target="_self" title="OSAFREE:def.23">Def23</a></i>;<br/>
<a NAME="E3:58_3_5"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">x1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">y</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">x2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">y</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="osafree.html#K20" title="OSAFREE:func.20">PTCongruence</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="msualg_4.html#K2" title="MSUALG_4:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">w3</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c23">y</font></span>)</span>
 
<b>by </b><i><a class="txt" href="osafree.html#E11:58_3"><i><font color="Green" title="E48">A68</font></i></a>, <a class="txt" href="osafree.html#E18:58_3"><i><font color="Green" title="E52">A88</font></i></a>, <a class="txt" href="osafree.html#E19:58_3"><i><font color="Green" title="E53">A89</font></i></a>, <a class="txt" href="osafree.html#E1:58_3_5"><i><font color="Green" title="E54">A90</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c24">x5</font>, <font color="Maroon" title="c25">y5</font> being    <a href="dtconstr.html#M1" title="DTCONSTR:mode.1">Element</a> of  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" title="OSAFREE:func.2">DTConOSA</a> <font color="Maroon" title="c2">X</font></span>)</span><b> such that </b><br/><a NAME="E5:58_3_5"/><i><font color="Green" title="E56">A92</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">x1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">y</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">x2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">y</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c24">x5</font>,<font color="Maroon" title="c25">y5</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>and </b><br/><a NAME="E6:58_3_5"/><i><font color="Green" title="E57">A93</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c24">x5</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">w3</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c23">y</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="osafree.html#K19" title="OSAFREE:func.19">PTClasses</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c25">y5</font>
 <b>by </b><i><a class="txt" href="osafree.html#E2:58_3_5"><i><font color="Green" title="E55">A91</font></i></a></i>;<br/>
<a NAME="E7:58_3_5"/><i><font color="Green" title="E58">A94</font></i>: 
( <font color="Maroon" title="c13">x1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">x5</font> &amp; <font color="Maroon" title="c14">x2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">y5</font> )
 
<b>by </b><i><a class="txt" href="osafree.html#E5:58_3_5"><i><font color="Green" title="E56">A92</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/>
<a NAME="E8:58_3_5"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c25">y5</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">w3</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c23">y</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="osafree.html#K19" title="OSAFREE:func.19">PTClasses</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c24">x5</font>
 
<b>by </b><i><a class="txt" href="osafree.html#E6:58_3_5"><i><font color="Green" title="E57">A93</font></i></a>, <a class="ref" href="osafree.html#T20" target="_self" title="OSAFREE:th.20">Th20</a></i>;<br/>
<b>hence </b><a NAME="E9:58_3_5"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">x2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">y</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">w3</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c23">y</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">y</font>
 <b>by </b><i><a class="txt" href="osafree.html#E1:58_3_5"><i><font color="Green" title="E54">A90</font></i></a>, <a class="txt" href="osafree.html#E7:58_3_5"><i><font color="Green" title="E58">A94</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E21:58_3"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c12">o2</font>,<span class="p4">(<span class="default"><a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">x2</font></span>)</span>,<font color="Maroon" title="c15">s3</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="osafree.html#K19" title="OSAFREE:func.19">PTClasses</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K9" title="OSAFREE:func.9">OSSym</a> <font color="Maroon" title="c11">o1</font>,<font color="Maroon" title="c2">X</font></span>)</span> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Maroon" title="c16">ts1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="osafree.html#E11:58_3"><i><font color="Green" title="E48">A68</font></i></a>, <a class="txt" href="osafree.html#E12:58_3"><i><font color="Green" title="E49">A69</font></i></a>, <a class="txt" href="osafree.html#E16:58_3"><i><font color="Green" title="E51">A87</font></i></a>, <a class="txt" href="osafree.html#E18:58_3"><i><font color="Green" title="E52">A88</font></i></a></i>;<br/>
<a NAME="E22:58_3"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c20">tx</font>,<font color="Maroon" title="c15">s3</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="osafree.html#K19" title="OSAFREE:func.19">PTClasses</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">ty</font>
 
<b>by </b><i><a class="txt" href="osafree.html#E7:58_3"><i><font color="Green" title="E45">A65</font></i></a>, <a class="ref" href="osafree.html#T20" target="_self" title="OSAFREE:th.20">Th20</a></i>;<br/>
<b>hence </b><a NAME="E23:58_3"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c11">o1</font>,<span class="p4">(<span class="default"><a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">x1</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c12">o2</font>,<span class="p4">(<span class="default"><a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">x2</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="osafree.html#K20" title="OSAFREE:func.20">PTCongruence</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="msualg_4.html#K2" title="MSUALG_4:func.2">.</a> <font color="Maroon" title="c15">s3</font>
 <b>by </b><i><a class="txt" href="osafree.html#E1:58_3"><i><font color="Green" title="E41">A61</font></i></a>, <a class="txt" href="osafree.html#E4:58_3"><i><font color="Green" title="E43">A63</font></i></a>, <a class="txt" href="osafree.html#E7:58_3"><i><font color="Green" title="E45">A65</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:58"/>
( <font color="Maroon" title="c3">R</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="osafree.html#K20" title="OSAFREE:func.20">PTCongruence</a> <font color="Maroon" title="c2">X</font> iff ( ( for <font color="Olive" title="b1">s1</font>, <font color="Olive" title="b2">s2</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font><br/> for <font color="Olive" title="b3">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">s1</font> holds <br/>( ( <font color="Olive" title="b1">s1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b2">s2</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="trees_4.html#K1" title="TREES_4:func.1">root-tree</a> <span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">x</font>,<font color="Olive" title="b1">s1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span>)</span>,<span class="p2">(<span class="default"><a href="trees_4.html#K1" title="TREES_4:func.1">root-tree</a> <span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">x</font>,<font color="Olive" title="b1">s1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">R</font> <a href="msualg_4.html#K1" title="MSUALG_4:func.1">.</a> <font color="Olive" title="b2">s2</font> ) &amp; ( for <font color="Olive" title="b4">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="trees_4.html#K1" title="TREES_4:func.1">root-tree</a> <span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">x</font>,<font color="Olive" title="b1">s1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span>)</span>,<font color="Olive" title="b4">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">R</font> <a href="msualg_4.html#K1" title="MSUALG_4:func.1">.</a> <font color="Olive" title="b2">s2</font> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b4">y</font>,<span class="p2">(<span class="default"><a href="trees_4.html#K1" title="TREES_4:func.1">root-tree</a> <span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">x</font>,<font color="Olive" title="b1">s1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">R</font> <a href="msualg_4.html#K1" title="MSUALG_4:func.1">.</a> <font color="Olive" title="b2">s2</font> ) holds <br/>( <font color="Olive" title="b1">s1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b2">s2</font> &amp; <font color="Olive" title="b4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="trees_4.html#K1" title="TREES_4:func.1">root-tree</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">x</font>,<font color="Olive" title="b1">s1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> ) ) ) ) &amp; ( for <font color="Olive" title="b1">o1</font>, <font color="Olive" title="b2">o2</font> being  <a href="osalg_1.html#NM2" title="OSALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">S</font><br/> for <font color="Olive" title="b3">x1</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Olive" title="b1">o1</font>,<span class="p1">(<span class="default"><a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</a> <font color="Maroon" title="c2">X</font></span>)</span><br/> for <font color="Olive" title="b4">x2</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Olive" title="b2">o2</font>,<span class="p1">(<span class="default"><a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</a> <font color="Maroon" title="c2">X</font></span>)</span><br/> for <font color="Olive" title="b5">s3</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font> holds <br/> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Olive" title="b1">o1</font>,<span class="p4">(<span class="default"><a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">x1</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Olive" title="b2">o2</font>,<span class="p4">(<span class="default"><a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b4">x2</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">R</font> <a href="msualg_4.html#K1" title="MSUALG_4:func.1">.</a> <font color="Olive" title="b5">s3</font> iff ( <font color="Olive" title="b1">o1</font> <a href="osalg_1.html#R1" title="OSALG_1:pred.1">~=</a> <font color="Olive" title="b2">o2</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Olive" title="b1">o1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Olive" title="b2">o2</font></span>)</span> &amp;  <a href="osalg_1.html#K1" title="OSALG_1:func.1">the_result_sort_of</a> <font color="Olive" title="b1">o1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b5">s3</font> &amp;  <a href="osalg_1.html#K1" title="OSALG_1:func.1">the_result_sort_of</a> <font color="Olive" title="b2">o2</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b5">s3</font> &amp;  ex <font color="Olive" title="b6">w3</font> being   <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  st <br/>(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Olive" title="b6">w3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Olive" title="b3">x1</font> &amp; ( for <font color="Olive" title="b7">y</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Olive" title="b6">w3</font> holds <br/><span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b3">x1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b7">y</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b4">x2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b7">y</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">R</font> <a href="msualg_4.html#K1" title="MSUALG_4:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b6">w3</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b7">y</font></span>)</span> ) ) ) ) ) ) )
 <b>by </b><i><a class="txt" href="osafree.html#E1:58"><i><font color="Green" title="E39">A1</font></i></a>, <a class="txt" href="osafree.html#E2:58"><i><font color="Green" title="E40">A47</font></i></a></i>;<br/>


</div>
